{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0001","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e88d0dd1032923de620ad8fb0ec3a4cba2ab95f3aa3e4c17594b62f334439e3f","family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137001,"semantic_template":"qf-density13-triangle-coordinate-qualification/[false,\"0\"]"},"fields":{"title":"Candidate Labeled Vertex Qualification and Signed Cross","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [0, 0], [1, 1]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.","answer":{"kind":"expression","value":"(No, 0, No)","explanation":"Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (No, 0, No)."},"skills":["Candidate Labeled Vertex Qualification and Signed Cross"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0002","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd211f5ffe656806d83fe78b48c6a41de4eb552a312233f3912f94e7b35f9d21","family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137002,"semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"0\"]"},"fields":{"title":"Candidate Labeled Vertex Qualification and Signed Cross","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 1], [2, 2]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.","answer":{"kind":"expression","value":"(Yes, 0, No)","explanation":"Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, 0, No)."},"skills":["Candidate Labeled Vertex Qualification and Signed Cross"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0003","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"931edf4ce6858cf9025864749e8f238394ceff28107d9adc4197925bfb885f5d","family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137003,"semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"-12\"]"},"fields":{"title":"Candidate Labeled Vertex Qualification and Signed Cross","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -12]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.","answer":{"kind":"expression","value":"(Yes, -12, Yes)","explanation":"Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, -12, Yes)."},"skills":["Candidate Labeled Vertex Qualification and Signed Cross"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0004","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c754c7463010e0b37d4a5fcce9d6f6306aae799c492d5592a724223ee04d6870","family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137004,"semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"-11\"]"},"fields":{"title":"Candidate Labeled Vertex Qualification and Signed Cross","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -11]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.","answer":{"kind":"expression","value":"(Yes, -11, Yes)","explanation":"Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, -11, Yes)."},"skills":["Candidate Labeled Vertex Qualification and Signed Cross"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0005","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"efde78c2e0bb8634dce204e555e1db7abff61db2c28dc287ce4ae42ef4c2d46f","family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137005,"semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"-10\"]"},"fields":{"title":"Candidate Labeled Vertex Qualification and Signed Cross","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -10]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.","answer":{"kind":"expression","value":"(Yes, -10, Yes)","explanation":"Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, -10, Yes)."},"skills":["Candidate Labeled Vertex Qualification and Signed Cross"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0006","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a48b5402f3edbd1cf9c5f9026a73d9894e93b70d8f8f05012b3a264dda67baa","family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137006,"semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"-9\"]"},"fields":{"title":"Candidate Labeled Vertex Qualification and Signed Cross","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -9]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.","answer":{"kind":"expression","value":"(Yes, -9, Yes)","explanation":"Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, -9, Yes)."},"skills":["Candidate Labeled Vertex Qualification and Signed Cross"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0007","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6cdd7bd062f5a941860b9ad51e84c6375075d4949e827a357532c15eb347b789","family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137007,"semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"-8\"]"},"fields":{"title":"Candidate Labeled Vertex Qualification and Signed Cross","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -8]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.","answer":{"kind":"expression","value":"(Yes, -8, Yes)","explanation":"Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, -8, Yes)."},"skills":["Candidate Labeled Vertex Qualification and Signed Cross"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0008","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c18087255c563a5b1f9d1c7a6cab5abb537688ff1cefb82f2241451c2e155942","family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137008,"semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"-7\"]"},"fields":{"title":"Candidate Labeled Vertex Qualification and Signed Cross","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -7]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.","answer":{"kind":"expression","value":"(Yes, -7, Yes)","explanation":"Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, -7, Yes)."},"skills":["Candidate Labeled Vertex Qualification and Signed Cross"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0009","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b76ad6f67eb61d76b8bf02c560d75103ceb646f2a09c3a9730b2f1aee85b2226","family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137009,"semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"-6\"]"},"fields":{"title":"Candidate Labeled Vertex Qualification and Signed Cross","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -6]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.","answer":{"kind":"expression","value":"(Yes, -6, Yes)","explanation":"Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, -6, Yes)."},"skills":["Candidate Labeled Vertex Qualification and Signed Cross"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0010","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30905b96369029b92d7622c7145e906b9ecc17d3a509be9d79c3cbff4658e361","family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137010,"semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"-5\"]"},"fields":{"title":"Candidate Labeled Vertex Qualification and Signed Cross","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -5]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.","answer":{"kind":"expression","value":"(Yes, -5, Yes)","explanation":"Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, -5, Yes)."},"skills":["Candidate Labeled Vertex Qualification and Signed Cross"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0011","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48825c5d4177377959b708b0800c697c83f65a25d528de048b66b38184ab51f5","family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137011,"semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"-4\"]"},"fields":{"title":"Candidate Labeled Vertex Qualification and Signed Cross","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -4]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.","answer":{"kind":"expression","value":"(Yes, -4, Yes)","explanation":"Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, -4, Yes)."},"skills":["Candidate Labeled Vertex Qualification and Signed Cross"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0012","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9178a465a60d799536b66aa9b5a1d8eeca8e859423d42c0397689c220f14e8d4","family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137012,"semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"-3\"]"},"fields":{"title":"Candidate Labeled Vertex Qualification and Signed Cross","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -3]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.","answer":{"kind":"expression","value":"(Yes, -3, Yes)","explanation":"Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, -3, Yes)."},"skills":["Candidate Labeled Vertex Qualification and Signed Cross"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0013","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8d2800920001c62475037dc0eab1dd81f5d1cfcc7a4daf291399a22a0a94aa7","family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137013,"semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"-2\"]"},"fields":{"title":"Candidate Labeled Vertex Qualification and Signed Cross","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -2]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.","answer":{"kind":"expression","value":"(Yes, -2, Yes)","explanation":"Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, -2, Yes)."},"skills":["Candidate Labeled Vertex Qualification and Signed Cross"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0014","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"89911ad57e7f0b63343b5d46167819671ef49412984fe334f1d5c5c1cc37a373","family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137014,"semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"-1\"]"},"fields":{"title":"Candidate Labeled Vertex Qualification and Signed Cross","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -1]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.","answer":{"kind":"expression","value":"(Yes, -1, Yes)","explanation":"Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, -1, Yes)."},"skills":["Candidate Labeled Vertex Qualification and Signed Cross"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0015","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1be82264bed3acea9af6dac21dc44a3166b3682f06084089387891dfeb8b37f","family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137015,"semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"1\"]"},"fields":{"title":"Candidate Labeled Vertex Qualification and Signed Cross","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 1]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.","answer":{"kind":"expression","value":"(Yes, 1, Yes)","explanation":"Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, 1, Yes)."},"skills":["Candidate Labeled Vertex Qualification and Signed Cross"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0016","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b326458887ff2a5996a3c2c4234111b2120e36bb8e4deb89a6a458da17d265d","family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137016,"semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"2\"]"},"fields":{"title":"Candidate Labeled Vertex Qualification and Signed Cross","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 2]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.","answer":{"kind":"expression","value":"(Yes, 2, Yes)","explanation":"Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, 2, Yes)."},"skills":["Candidate Labeled Vertex Qualification and Signed Cross"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0017","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74e8b479da9ff3c38d1fb29aa410d05f41bdc877f9edd4aa8abaf70a22ec623b","family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137017,"semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"3\"]"},"fields":{"title":"Candidate Labeled Vertex Qualification and Signed Cross","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 3]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.","answer":{"kind":"expression","value":"(Yes, 3, Yes)","explanation":"Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, 3, Yes)."},"skills":["Candidate Labeled Vertex Qualification and Signed Cross"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0018","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a45d7eac0ed60f805c90e3a0d6da66817e3bc51bfadcd92f31993df5f57a2af","family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137018,"semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"4\"]"},"fields":{"title":"Candidate Labeled Vertex Qualification and Signed Cross","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 4]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.","answer":{"kind":"expression","value":"(Yes, 4, Yes)","explanation":"Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, 4, Yes)."},"skills":["Candidate Labeled Vertex Qualification and Signed Cross"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0019","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff62b727c7d2879eee8bc351032cca76863200f405b8fc45e4e26306f6ff58ae","family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137019,"semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"5\"]"},"fields":{"title":"Candidate Labeled Vertex Qualification and Signed Cross","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 5]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.","answer":{"kind":"expression","value":"(Yes, 5, Yes)","explanation":"Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, 5, Yes)."},"skills":["Candidate Labeled Vertex Qualification and Signed Cross"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0020","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"db5d65bc9a89028abc3d79dc6bb805c99d2fce1f8c548970ab53d68f81f0445c","family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137020,"semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"6\"]"},"fields":{"title":"Candidate Labeled Vertex Qualification and Signed Cross","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 6]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.","answer":{"kind":"expression","value":"(Yes, 6, Yes)","explanation":"Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, 6, Yes)."},"skills":["Candidate Labeled Vertex Qualification and Signed Cross"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0021","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7e20a5d7c40f8bd90f020b9c257d95ab8f05a2675ff418ed8012aab1d703fe2","family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137021,"semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"7\"]"},"fields":{"title":"Candidate Labeled Vertex Qualification and Signed Cross","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 7]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.","answer":{"kind":"expression","value":"(Yes, 7, Yes)","explanation":"Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, 7, Yes)."},"skills":["Candidate Labeled Vertex Qualification and Signed Cross"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0022","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"97769bb6919b0b15f37db20ae2f34e86ce3485244219edf150acd7cd4cf2d6c8","family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137022,"semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"8\"]"},"fields":{"title":"Candidate Labeled Vertex Qualification and Signed Cross","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 8]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.","answer":{"kind":"expression","value":"(Yes, 8, Yes)","explanation":"Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, 8, Yes)."},"skills":["Candidate Labeled Vertex Qualification and Signed Cross"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0023","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3483e15a4d9a050c9924c9f7cb1536b48736727e089fd0f6fdfe567687b63cb","family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137023,"semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"9\"]"},"fields":{"title":"Candidate Labeled Vertex Qualification and Signed Cross","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 9]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.","answer":{"kind":"expression","value":"(Yes, 9, Yes)","explanation":"Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, 9, Yes)."},"skills":["Candidate Labeled Vertex Qualification and Signed Cross"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0024","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa63e4b5e7ab94842a2a84817e74a47ce8f413f55b1475dca4c77e131a4497c3","family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137024,"semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"10\"]"},"fields":{"title":"Candidate Labeled Vertex Qualification and Signed Cross","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 10]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.","answer":{"kind":"expression","value":"(Yes, 10, Yes)","explanation":"Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, 10, Yes)."},"skills":["Candidate Labeled Vertex Qualification and Signed Cross"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0025","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1cdfb0543db312104b5111a327520bbf3feca6197b6c7bfd9cea926734fb60c7","family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137025,"semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"11\"]"},"fields":{"title":"Candidate Labeled Vertex Qualification and Signed Cross","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 11]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.","answer":{"kind":"expression","value":"(Yes, 11, Yes)","explanation":"Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, 11, Yes)."},"skills":["Candidate Labeled Vertex Qualification and Signed Cross"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0026","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d84dfd6290ea6fd624fbfa9694c72a099ed17453c06ac85089972a98a3049960","family_id":"qf-density13-triangle-opposite-side-squares","seed":202610137026,"semantic_template":"qf-density13-triangle-opposite-side-squares/[1,\"1\"]"},"fields":{"title":"Labeled Opposite Endpoints and Squared Side","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 1], [0, 0], [1, 0]]. The triangle is qualified as distinct and noncollinear, vertex=1. Give (other two vertex indices, opposite side’s squared L2 distance), not its distance.","answer":{"kind":"expression","value":"([2, 3], 1)","explanation":"Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([2, 3], 1)."},"skills":["Labeled Opposite Endpoints and Squared Side"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0027","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f99369dfb503173f658f5ddc5a173aa4220dcf34495f15cf7c2bff774e57f00","family_id":"qf-density13-triangle-opposite-side-squares","seed":202610137027,"semantic_template":"qf-density13-triangle-opposite-side-squares/[2,\"4\"]"},"fields":{"title":"Labeled Opposite Endpoints and Squared Side","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [0, 1], [2, 0]]. The triangle is qualified as distinct and noncollinear, vertex=2. Give (other two vertex indices, opposite side’s squared L2 distance), not its distance.","answer":{"kind":"expression","value":"([1, 3], 4)","explanation":"Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([1, 3], 4)."},"skills":["Labeled Opposite Endpoints and Squared Side"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0028","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"663b1b30a40bab43c45cf5e1dc7fb3a7b0b32d1752cc53a81bc02c27a83c9a65","family_id":"qf-density13-triangle-opposite-side-squares","seed":202610137028,"semantic_template":"qf-density13-triangle-opposite-side-squares/[3,\"9\"]"},"fields":{"title":"Labeled Opposite Endpoints and Squared Side","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [3, 0], [0, 1]]. The triangle is qualified as distinct and noncollinear, vertex=3. Give (other two vertex indices, opposite side’s squared L2 distance), not its distance.","answer":{"kind":"expression","value":"([1, 2], 9)","explanation":"Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([1, 2], 9)."},"skills":["Labeled Opposite Endpoints and Squared Side"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0029","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16c9133b793979342079bc384a106539a59c54e6d766ff74935102d93cd42503","family_id":"qf-density13-triangle-opposite-side-squares","seed":202610137029,"semantic_template":"qf-density13-triangle-opposite-side-squares/[1,\"16\"]"},"fields":{"title":"Labeled Opposite Endpoints and Squared Side","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 1], [0, 0], [4, 0]]. The triangle is qualified as distinct and noncollinear, vertex=1. Give (other two vertex indices, opposite side’s squared L2 distance), not its distance.","answer":{"kind":"expression","value":"([2, 3], 16)","explanation":"Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([2, 3], 16)."},"skills":["Labeled Opposite Endpoints and Squared Side"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0030","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8088df6886a6165e7a560d1a23b867b4f0ac41ead5ffbcec69957926e41fef01","family_id":"qf-density13-triangle-opposite-side-squares","seed":202610137030,"semantic_template":"qf-density13-triangle-opposite-side-squares/[2,\"25\"]"},"fields":{"title":"Labeled Opposite Endpoints and Squared Side","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [0, 1], [5, 0]]. The triangle is qualified as distinct and noncollinear, vertex=2. Give (other two vertex indices, opposite side’s squared L2 distance), not its distance.","answer":{"kind":"expression","value":"([1, 3], 25)","explanation":"Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([1, 3], 25)."},"skills":["Labeled Opposite Endpoints and Squared Side"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0031","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6d5f705113cc91fd54cdad2cb73e81a468e2fddbeae1b6e28cead86cf9d2649","family_id":"qf-density13-triangle-opposite-side-squares","seed":202610137031,"semantic_template":"qf-density13-triangle-opposite-side-squares/[3,\"36\"]"},"fields":{"title":"Labeled Opposite Endpoints and Squared Side","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [6, 0], [0, 1]]. The triangle is qualified as distinct and noncollinear, vertex=3. Give (other two vertex indices, opposite side’s squared L2 distance), not its distance.","answer":{"kind":"expression","value":"([1, 2], 36)","explanation":"Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([1, 2], 36)."},"skills":["Labeled Opposite Endpoints and Squared Side"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0032","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"79ff998847062aa58b88953136010b4e59ead61b7ce265eb1cf289efb26c19da","family_id":"qf-density13-triangle-opposite-side-squares","seed":202610137032,"semantic_template":"qf-density13-triangle-opposite-side-squares/[1,\"49\"]"},"fields":{"title":"Labeled Opposite Endpoints and Squared Side","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 1], [0, 0], [7, 0]]. The triangle is qualified as distinct and noncollinear, vertex=1. Give (other two vertex indices, opposite side’s squared L2 distance), not its distance.","answer":{"kind":"expression","value":"([2, 3], 49)","explanation":"Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([2, 3], 49)."},"skills":["Labeled Opposite Endpoints and Squared Side"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0033","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c554c4797bf6f30be4e759befeca97d281eeb41d4e1157d0f5500466fa838d3","family_id":"qf-density13-triangle-opposite-side-squares","seed":202610137033,"semantic_template":"qf-density13-triangle-opposite-side-squares/[2,\"64\"]"},"fields":{"title":"Labeled Opposite Endpoints and Squared Side","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [0, 1], [8, 0]]. The triangle is qualified as distinct and noncollinear, vertex=2. Give (other two vertex indices, opposite side’s squared L2 distance), not its distance.","answer":{"kind":"expression","value":"([1, 3], 64)","explanation":"Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([1, 3], 64)."},"skills":["Labeled Opposite Endpoints and Squared Side"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0034","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17354745ffde836a07aa643df6f4aa74eb7892eeaa8baed935485a8fc070c23e","family_id":"qf-density13-triangle-opposite-side-squares","seed":202610137034,"semantic_template":"qf-density13-triangle-opposite-side-squares/[3,\"81\"]"},"fields":{"title":"Labeled Opposite Endpoints and Squared Side","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [9, 0], [0, 1]]. The triangle is qualified as distinct and noncollinear, vertex=3. Give (other two vertex indices, opposite side’s squared L2 distance), not its distance.","answer":{"kind":"expression","value":"([1, 2], 81)","explanation":"Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([1, 2], 81)."},"skills":["Labeled Opposite Endpoints and Squared Side"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0035","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5fc3b0b7bfbe9cebcaacc496f89f17c5d953abbd6fb1b5ac0d7e50becbb036a6","family_id":"qf-density13-triangle-opposite-side-squares","seed":202610137035,"semantic_template":"qf-density13-triangle-opposite-side-squares/[1,\"100\"]"},"fields":{"title":"Labeled Opposite Endpoints and Squared Side","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 1], [0, 0], [10, 0]]. The triangle is qualified as distinct and noncollinear, vertex=1. Give (other two vertex indices, opposite side’s squared L2 distance), not its distance.","answer":{"kind":"expression","value":"([2, 3], 100)","explanation":"Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([2, 3], 100)."},"skills":["Labeled Opposite Endpoints and Squared Side"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0036","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69eb124b929c2666a2a5e532a9fcd72c2155998ec2fe34f6d28d45ee047d2a4a","family_id":"qf-density13-triangle-opposite-side-squares","seed":202610137036,"semantic_template":"qf-density13-triangle-opposite-side-squares/[2,\"121\"]"},"fields":{"title":"Labeled Opposite Endpoints and Squared Side","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [0, 1], [11, 0]]. The triangle is qualified as distinct and noncollinear, vertex=2. Give (other two vertex indices, opposite side’s squared L2 distance), not its distance.","answer":{"kind":"expression","value":"([1, 3], 121)","explanation":"Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([1, 3], 121)."},"skills":["Labeled Opposite Endpoints and Squared Side"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0037","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b5b5a3681d951476a8fba259e4af5b9c334e8b362509b53f12354c5f1a302aaa","family_id":"qf-density13-triangle-opposite-side-squares","seed":202610137037,"semantic_template":"qf-density13-triangle-opposite-side-squares/[3,\"144\"]"},"fields":{"title":"Labeled Opposite Endpoints and Squared Side","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [12, 0], [0, 1]]. The triangle is qualified as distinct and noncollinear, vertex=3. Give (other two vertex indices, opposite side’s squared L2 distance), not its distance.","answer":{"kind":"expression","value":"([1, 2], 144)","explanation":"Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([1, 2], 144)."},"skills":["Labeled Opposite Endpoints and Squared Side"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0038","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"beffd5f838d4d032b5d52c1d18395b4bd3837a741e6544ea851e400605367eba","family_id":"qf-density13-triangle-opposite-side-squares","seed":202610137038,"semantic_template":"qf-density13-triangle-opposite-side-squares/[1,\"169\"]"},"fields":{"title":"Labeled Opposite Endpoints and Squared Side","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 1], [0, 0], [13, 0]]. The triangle is qualified as distinct and noncollinear, vertex=1. Give (other two vertex indices, opposite side’s squared L2 distance), not its distance.","answer":{"kind":"expression","value":"([2, 3], 169)","explanation":"Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([2, 3], 169)."},"skills":["Labeled Opposite Endpoints and Squared Side"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0039","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"15a8bf688ed5a77e552383069950fe43ab646bf71ba2181a091a701a321a62c7","family_id":"qf-density13-triangle-opposite-side-squares","seed":202610137039,"semantic_template":"qf-density13-triangle-opposite-side-squares/[2,\"196\"]"},"fields":{"title":"Labeled Opposite Endpoints and Squared Side","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [0, 1], [14, 0]]. The triangle is qualified as distinct and noncollinear, vertex=2. Give (other two vertex indices, opposite side’s squared L2 distance), not its distance.","answer":{"kind":"expression","value":"([1, 3], 196)","explanation":"Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([1, 3], 196)."},"skills":["Labeled Opposite Endpoints and Squared Side"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0040","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8da0b5633e7b4926c14aae253951f8b1a2f1364f4074b8b8fae337a17ee7c012","family_id":"qf-density13-triangle-opposite-side-squares","seed":202610137040,"semantic_template":"qf-density13-triangle-opposite-side-squares/[3,\"225\"]"},"fields":{"title":"Labeled Opposite Endpoints and Squared Side","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [15, 0], [0, 1]]. The triangle is qualified as distinct and noncollinear, vertex=3. Give (other two vertex indices, opposite side’s squared L2 distance), not its distance.","answer":{"kind":"expression","value":"([1, 2], 225)","explanation":"Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([1, 2], 225)."},"skills":["Labeled Opposite Endpoints and Squared Side"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0041","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"771e8823a385fb1f526ce6a9330e74a1c142b337b27dc9ba219b71081827fb5c","family_id":"qf-density13-triangle-opposite-side-squares","seed":202610137041,"semantic_template":"qf-density13-triangle-opposite-side-squares/[1,\"256\"]"},"fields":{"title":"Labeled Opposite Endpoints and Squared Side","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 1], [0, 0], [16, 0]]. The triangle is qualified as distinct and noncollinear, vertex=1. Give (other two vertex indices, opposite side’s squared L2 distance), not its distance.","answer":{"kind":"expression","value":"([2, 3], 256)","explanation":"Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([2, 3], 256)."},"skills":["Labeled Opposite Endpoints and Squared Side"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0042","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7e0507c3cae15466cfec127065916f3ce07cd4d22115cb4f174c9651689bc32","family_id":"qf-density13-triangle-opposite-side-squares","seed":202610137042,"semantic_template":"qf-density13-triangle-opposite-side-squares/[2,\"289\"]"},"fields":{"title":"Labeled Opposite Endpoints and Squared Side","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [0, 1], [17, 0]]. The triangle is qualified as distinct and noncollinear, vertex=2. Give (other two vertex indices, opposite side’s squared L2 distance), not its distance.","answer":{"kind":"expression","value":"([1, 3], 289)","explanation":"Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([1, 3], 289)."},"skills":["Labeled Opposite Endpoints and Squared Side"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0043","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"03c8a846a10e2acf9c11b466bcae151a3083075183661b0e0ca26a0b4612691e","family_id":"qf-density13-triangle-opposite-side-squares","seed":202610137043,"semantic_template":"qf-density13-triangle-opposite-side-squares/[3,\"324\"]"},"fields":{"title":"Labeled Opposite Endpoints and Squared Side","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [18, 0], [0, 1]]. The triangle is qualified as distinct and noncollinear, vertex=3. Give (other two vertex indices, opposite side’s squared L2 distance), not its distance.","answer":{"kind":"expression","value":"([1, 2], 324)","explanation":"Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([1, 2], 324)."},"skills":["Labeled Opposite Endpoints and Squared Side"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0044","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1288bf59fb84a2aa21d4ecc92bd8536b7fdc1f63f8d26b5285d53056d2250b54","family_id":"qf-density13-triangle-opposite-side-squares","seed":202610137044,"semantic_template":"qf-density13-triangle-opposite-side-squares/[1,\"361\"]"},"fields":{"title":"Labeled Opposite Endpoints and Squared Side","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 1], [0, 0], [19, 0]]. The triangle is qualified as distinct and noncollinear, vertex=1. Give (other two vertex indices, opposite side’s squared L2 distance), not its distance.","answer":{"kind":"expression","value":"([2, 3], 361)","explanation":"Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([2, 3], 361)."},"skills":["Labeled Opposite Endpoints and Squared Side"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0045","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd27ff583bc645f8cb71fcf64fdcc990b3ccbc7daba93597ab7dd99cac9dac92","family_id":"qf-density13-triangle-opposite-side-squares","seed":202610137045,"semantic_template":"qf-density13-triangle-opposite-side-squares/[2,\"400\"]"},"fields":{"title":"Labeled Opposite Endpoints and Squared Side","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [0, 1], [20, 0]]. The triangle is qualified as distinct and noncollinear, vertex=2. Give (other two vertex indices, opposite side’s squared L2 distance), not its distance.","answer":{"kind":"expression","value":"([1, 3], 400)","explanation":"Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([1, 3], 400)."},"skills":["Labeled Opposite Endpoints and Squared Side"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0046","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"15ecbd6ddb6cbba1610fc1ee4abc396983ce8855e37616b5905a45bbb6ac6421","family_id":"qf-density13-triangle-opposite-side-squares","seed":202610137046,"semantic_template":"qf-density13-triangle-opposite-side-squares/[3,\"441\"]"},"fields":{"title":"Labeled Opposite Endpoints and Squared Side","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [21, 0], [0, 1]]. The triangle is qualified as distinct and noncollinear, vertex=3. Give (other two vertex indices, opposite side’s squared L2 distance), not its distance.","answer":{"kind":"expression","value":"([1, 2], 441)","explanation":"Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([1, 2], 441)."},"skills":["Labeled Opposite Endpoints and Squared Side"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0047","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f7f2da52ceb9ccff1d56aa244442e0fb09b55d0cff9a01f2ed386a899d6d540","family_id":"qf-density13-triangle-opposite-side-squares","seed":202610137047,"semantic_template":"qf-density13-triangle-opposite-side-squares/[1,\"484\"]"},"fields":{"title":"Labeled Opposite Endpoints and Squared Side","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 1], [0, 0], [22, 0]]. The triangle is qualified as distinct and noncollinear, vertex=1. Give (other two vertex indices, opposite side’s squared L2 distance), not its distance.","answer":{"kind":"expression","value":"([2, 3], 484)","explanation":"Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([2, 3], 484)."},"skills":["Labeled Opposite Endpoints and Squared Side"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0048","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91c1151c2079671058722efb7ee4b48d0a75e0447354c456e31cc3b22424927d","family_id":"qf-density13-triangle-opposite-side-squares","seed":202610137048,"semantic_template":"qf-density13-triangle-opposite-side-squares/[2,\"529\"]"},"fields":{"title":"Labeled Opposite Endpoints and Squared Side","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [0, 1], [23, 0]]. The triangle is qualified as distinct and noncollinear, vertex=2. Give (other two vertex indices, opposite side’s squared L2 distance), not its distance.","answer":{"kind":"expression","value":"([1, 3], 529)","explanation":"Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([1, 3], 529)."},"skills":["Labeled Opposite Endpoints and Squared Side"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0049","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67e2f563f0d0b0e58d40c8095971962391f22fbe427b55ee5d894d4e26a28f0d","family_id":"qf-density13-triangle-opposite-side-squares","seed":202610137049,"semantic_template":"qf-density13-triangle-opposite-side-squares/[3,\"576\"]"},"fields":{"title":"Labeled Opposite Endpoints and Squared Side","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [24, 0], [0, 1]]. The triangle is qualified as distinct and noncollinear, vertex=3. Give (other two vertex indices, opposite side’s squared L2 distance), not its distance.","answer":{"kind":"expression","value":"([1, 2], 576)","explanation":"Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([1, 2], 576)."},"skills":["Labeled Opposite Endpoints and Squared Side"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0050","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae68007a0164ec6e418d8597f8cef186d34180a8ea5828982931d372a6f0df6c","family_id":"qf-density13-triangle-opposite-side-squares","seed":202610137050,"semantic_template":"qf-density13-triangle-opposite-side-squares/[1,\"625\"]"},"fields":{"title":"Labeled Opposite Endpoints and Squared Side","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 1], [0, 0], [25, 0]]. The triangle is qualified as distinct and noncollinear, vertex=1. Give (other two vertex indices, opposite side’s squared L2 distance), not its distance.","answer":{"kind":"expression","value":"([2, 3], 625)","explanation":"Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([2, 3], 625)."},"skills":["Labeled Opposite Endpoints and Squared Side"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0051","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c35b54a85bf7c865d8bbfc0ab12c161bb34db7cd6e1d9e8dd56924a773492893","family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137051,"semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"-12\""},"fields":{"title":"Cyclic versus Swapped Signed Coordinate Crosses","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -12]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.","answer":{"kind":"expression","value":"(-12, -12, 12)","explanation":"Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (-12, -12, 12)."},"skills":["Cyclic versus Swapped Signed Coordinate Crosses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0052","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39d70b129d59668dca68f5d68ca50a1aee011b04b799b9be3fbcc0622fb12d06","family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137052,"semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"-11\""},"fields":{"title":"Cyclic versus Swapped Signed Coordinate Crosses","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -11]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.","answer":{"kind":"expression","value":"(-11, -11, 11)","explanation":"Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (-11, -11, 11)."},"skills":["Cyclic versus Swapped Signed Coordinate Crosses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0053","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d410c85a7011e33ff325473a5060547f8ea8755fb6ab061c12d7508d16271341","family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137053,"semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"-10\""},"fields":{"title":"Cyclic versus Swapped Signed Coordinate Crosses","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -10]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.","answer":{"kind":"expression","value":"(-10, -10, 10)","explanation":"Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (-10, -10, 10)."},"skills":["Cyclic versus Swapped Signed Coordinate Crosses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0054","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a8693c1fd41d9130861e96a1bead313e903431c59b1f5fb01e4aa3dbf156147","family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137054,"semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"-9\""},"fields":{"title":"Cyclic versus Swapped Signed Coordinate Crosses","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -9]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.","answer":{"kind":"expression","value":"(-9, -9, 9)","explanation":"Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (-9, -9, 9)."},"skills":["Cyclic versus Swapped Signed Coordinate Crosses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0055","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5fa1867ebd4ca9f22e3c49be42289d9e6f7e2dfa41608df4f2f7203ec68a673","family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137055,"semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"-8\""},"fields":{"title":"Cyclic versus Swapped Signed Coordinate Crosses","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -8]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.","answer":{"kind":"expression","value":"(-8, -8, 8)","explanation":"Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (-8, -8, 8)."},"skills":["Cyclic versus Swapped Signed Coordinate Crosses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0056","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf80041349596e8b9ac6596d8eb26bacf440a35c009e1e5fbdb356c3ee814bfa","family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137056,"semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"-7\""},"fields":{"title":"Cyclic versus Swapped Signed Coordinate Crosses","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -7]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.","answer":{"kind":"expression","value":"(-7, -7, 7)","explanation":"Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (-7, -7, 7)."},"skills":["Cyclic versus Swapped Signed Coordinate Crosses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0057","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30cb5d8a810f4cfb5543e232a12a430c32885d4b2fb03118b1959af9a8447d44","family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137057,"semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"-6\""},"fields":{"title":"Cyclic versus Swapped Signed Coordinate Crosses","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -6]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.","answer":{"kind":"expression","value":"(-6, -6, 6)","explanation":"Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (-6, -6, 6)."},"skills":["Cyclic versus Swapped Signed Coordinate Crosses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0058","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"340efe0bfc12b69761205cd7f5f5b07754f9b943aa82407744c811dea0cd8d2e","family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137058,"semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"-5\""},"fields":{"title":"Cyclic versus Swapped Signed Coordinate Crosses","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -5]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.","answer":{"kind":"expression","value":"(-5, -5, 5)","explanation":"Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (-5, -5, 5)."},"skills":["Cyclic versus Swapped Signed Coordinate Crosses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0059","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"29def3aa2fd98af39baab2b918c4b3ecab2aa444ddedbc957f4c03cf8cca8c03","family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137059,"semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"-4\""},"fields":{"title":"Cyclic versus Swapped Signed Coordinate Crosses","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -4]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.","answer":{"kind":"expression","value":"(-4, -4, 4)","explanation":"Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (-4, -4, 4)."},"skills":["Cyclic versus Swapped Signed Coordinate Crosses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0060","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cecfec2d890872d6181085a83479e0e77e7f60ad3fd3305f865d71580c5aa5e2","family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137060,"semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"-3\""},"fields":{"title":"Cyclic versus Swapped Signed Coordinate Crosses","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -3]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.","answer":{"kind":"expression","value":"(-3, -3, 3)","explanation":"Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (-3, -3, 3)."},"skills":["Cyclic versus Swapped Signed Coordinate Crosses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0061","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c7095d9a88127133d972c4f9da8916b78615d4099cf7f0c0232ff3eff19de323","family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137061,"semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"-2\""},"fields":{"title":"Cyclic versus Swapped Signed Coordinate Crosses","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -2]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.","answer":{"kind":"expression","value":"(-2, -2, 2)","explanation":"Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (-2, -2, 2)."},"skills":["Cyclic versus Swapped Signed Coordinate Crosses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0062","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f826249278d55cbabfcd9322bb50d3551b163640be8fb84be6701f5a9220edf1","family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137062,"semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"-1\""},"fields":{"title":"Cyclic versus Swapped Signed Coordinate Crosses","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -1]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.","answer":{"kind":"expression","value":"(-1, -1, 1)","explanation":"Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (-1, -1, 1)."},"skills":["Cyclic versus Swapped Signed Coordinate Crosses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0063","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1be2d10afaef7625eb399dbf1569d645c0fdec429ed999552ad7e6801e24899d","family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137063,"semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"1\""},"fields":{"title":"Cyclic versus Swapped Signed Coordinate Crosses","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 1]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.","answer":{"kind":"expression","value":"(1, 1, -1)","explanation":"Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1, 1, -1)."},"skills":["Cyclic versus Swapped Signed Coordinate Crosses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0064","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9411c4c92008423e52c1cd388ab5adc3b0408edfc85e8cdda36546824a4a1e5a","family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137064,"semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"2\""},"fields":{"title":"Cyclic versus Swapped Signed Coordinate Crosses","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 2]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.","answer":{"kind":"expression","value":"(2, 2, -2)","explanation":"Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (2, 2, -2)."},"skills":["Cyclic versus Swapped Signed Coordinate Crosses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0065","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1edadc49f591a69d08c181a225d35956aaedb6095ed92ce49a5bb76b974190eb","family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137065,"semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"3\""},"fields":{"title":"Cyclic versus Swapped Signed Coordinate Crosses","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 3]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.","answer":{"kind":"expression","value":"(3, 3, -3)","explanation":"Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (3, 3, -3)."},"skills":["Cyclic versus Swapped Signed Coordinate Crosses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0066","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3751e497a341e72ae8e2708c5b21de316aedf865a1c5103207a22b8f9041812e","family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137066,"semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"4\""},"fields":{"title":"Cyclic versus Swapped Signed Coordinate Crosses","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 4]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.","answer":{"kind":"expression","value":"(4, 4, -4)","explanation":"Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (4, 4, -4)."},"skills":["Cyclic versus Swapped Signed Coordinate Crosses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0067","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b9428c572a8d250e363e30faa0bde4e9a2fe1c980cfc6ce986b532122767b08f","family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137067,"semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"5\""},"fields":{"title":"Cyclic versus Swapped Signed Coordinate Crosses","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 5]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.","answer":{"kind":"expression","value":"(5, 5, -5)","explanation":"Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (5, 5, -5)."},"skills":["Cyclic versus Swapped Signed Coordinate Crosses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0068","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d0dbc082c515b2e7e293c683843d42c920ae8c59c3bcb6c2f62aa43b825cecf","family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137068,"semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"6\""},"fields":{"title":"Cyclic versus Swapped Signed Coordinate Crosses","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 6]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.","answer":{"kind":"expression","value":"(6, 6, -6)","explanation":"Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (6, 6, -6)."},"skills":["Cyclic versus Swapped Signed Coordinate Crosses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0069","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2fc829077a904d3109e3600092d81982438095e8351681468a16c56c84ddd083","family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137069,"semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"7\""},"fields":{"title":"Cyclic versus Swapped Signed Coordinate Crosses","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 7]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.","answer":{"kind":"expression","value":"(7, 7, -7)","explanation":"Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (7, 7, -7)."},"skills":["Cyclic versus Swapped Signed Coordinate Crosses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0070","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3b46c8855ffd1b5241dc195ba48a092dd99c6affb4686ca5d60b3db03fabcb3","family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137070,"semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"8\""},"fields":{"title":"Cyclic versus Swapped Signed Coordinate Crosses","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 8]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.","answer":{"kind":"expression","value":"(8, 8, -8)","explanation":"Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (8, 8, -8)."},"skills":["Cyclic versus Swapped Signed Coordinate Crosses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0071","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e0f700f21abfbba7b2bb113104a7097851dd7c373d6e40c507cd86f2c47f7d3","family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137071,"semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"9\""},"fields":{"title":"Cyclic versus Swapped Signed Coordinate Crosses","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 9]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.","answer":{"kind":"expression","value":"(9, 9, -9)","explanation":"Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (9, 9, -9)."},"skills":["Cyclic versus Swapped Signed Coordinate Crosses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0072","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19ce3fdb7ad86b35c660805e0914c0a9e097c129fc1d56745be4296557bf0b6f","family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137072,"semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"10\""},"fields":{"title":"Cyclic versus Swapped Signed Coordinate Crosses","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 10]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.","answer":{"kind":"expression","value":"(10, 10, -10)","explanation":"Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (10, 10, -10)."},"skills":["Cyclic versus Swapped Signed Coordinate Crosses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0073","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ec8215969a3c9016d4f6b29e012404919383b075af9fe0e3fb9163390efc2db","family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137073,"semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"11\""},"fields":{"title":"Cyclic versus Swapped Signed Coordinate Crosses","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 11]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.","answer":{"kind":"expression","value":"(11, 11, -11)","explanation":"Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (11, 11, -11)."},"skills":["Cyclic versus Swapped Signed Coordinate Crosses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0074","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"33c5b56a6623765f1484db3d9101832bbdbac30d0f0c06081612899efd469a4c","family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137074,"semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"12\""},"fields":{"title":"Cyclic versus Swapped Signed Coordinate Crosses","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 12]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.","answer":{"kind":"expression","value":"(12, 12, -12)","explanation":"Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (12, 12, -12)."},"skills":["Cyclic versus Swapped Signed Coordinate Crosses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0075","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"487e8e1b4cb2ef5c4f949327be5ba0f5ec8c5d0080ea24b0b161c77ae5ac753b","family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137075,"semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"13\""},"fields":{"title":"Cyclic versus Swapped Signed Coordinate Crosses","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 13]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.","answer":{"kind":"expression","value":"(13, 13, -13)","explanation":"Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (13, 13, -13)."},"skills":["Cyclic versus Swapped Signed Coordinate Crosses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0076","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9cb66a182610ab5f47a08ab1df2819adbe033e9be974acd5153218c7b3750b0d","family_id":"qf-density13-triangle-median-equality","seed":202610137076,"semantic_template":"qf-density13-triangle-median-equality/[1,[\"1/2\",\"0/1\"],\"1/4\",\"5/4\"]"},"fields":{"title":"Opposite Midpoint and Three Squared Segments","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 1], [0, 0], [1, 0]]. The points are distinct and noncollinear, vertex=1, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 0/1, 1/4, 1/4, 5/4)","explanation":"Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 1/4, 1/4, 5/4)."},"skills":["Opposite Midpoint and Three Squared Segments"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0077","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c173bb2b72c9917bf0f3c4350ad436c9c69aab65db012938a290f5fb3ca7db9","family_id":"qf-density13-triangle-median-equality","seed":202610137077,"semantic_template":"qf-density13-triangle-median-equality/[2,[\"1/2\",\"0/1\"],\"9/4\",\"17/4\"]"},"fields":{"title":"Opposite Midpoint and Three Squared Segments","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[-1, 0], [0, 2], [2, 0]]. The points are distinct and noncollinear, vertex=2, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 0/1, 9/4, 9/4, 17/4)","explanation":"Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 9/4, 9/4, 17/4)."},"skills":["Opposite Midpoint and Three Squared Segments"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0078","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c28e4134739b17fac0056dbd7b6ba48b857a6264cd0a9b27868afa4162361df7","family_id":"qf-density13-triangle-median-equality","seed":202610137078,"semantic_template":"qf-density13-triangle-median-equality/[3,[\"1/2\",\"0/1\"],\"25/4\",\"37/4\"]"},"fields":{"title":"Opposite Midpoint and Three Squared Segments","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[-2, 0], [3, 0], [0, 3]]. The points are distinct and noncollinear, vertex=3, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 0/1, 25/4, 25/4, 37/4)","explanation":"Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 25/4, 25/4, 37/4)."},"skills":["Opposite Midpoint and Three Squared Segments"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0079","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7554dd345e70917a524b3d38e5ae1d284c2cc3405c809c91096a69fecbeee403","family_id":"qf-density13-triangle-median-equality","seed":202610137079,"semantic_template":"qf-density13-triangle-median-equality/[1,[\"1/2\",\"0/1\"],\"49/4\",\"65/4\"]"},"fields":{"title":"Opposite Midpoint and Three Squared Segments","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 4], [-3, 0], [4, 0]]. The points are distinct and noncollinear, vertex=1, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 0/1, 49/4, 49/4, 65/4)","explanation":"Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 49/4, 49/4, 65/4)."},"skills":["Opposite Midpoint and Three Squared Segments"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0080","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d4653a5692222d558fc31673cc3962f6a1492757a615c20d2e835446cbc5254","family_id":"qf-density13-triangle-median-equality","seed":202610137080,"semantic_template":"qf-density13-triangle-median-equality/[2,[\"1/2\",\"0/1\"],\"81/4\",\"101/4\"]"},"fields":{"title":"Opposite Midpoint and Three Squared Segments","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[-4, 0], [0, 5], [5, 0]]. The points are distinct and noncollinear, vertex=2, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 0/1, 81/4, 81/4, 101/4)","explanation":"Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 81/4, 81/4, 101/4)."},"skills":["Opposite Midpoint and Three Squared Segments"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0081","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48aa3f7dbeaab1c00ee986f8044a1d9df7d9ba783643f3eaffe41372e113c957","family_id":"qf-density13-triangle-median-equality","seed":202610137081,"semantic_template":"qf-density13-triangle-median-equality/[3,[\"1/2\",\"0/1\"],\"121/4\",\"145/4\"]"},"fields":{"title":"Opposite Midpoint and Three Squared Segments","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[-5, 0], [6, 0], [0, 6]]. The points are distinct and noncollinear, vertex=3, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 0/1, 121/4, 121/4, 145/4)","explanation":"Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 121/4, 121/4, 145/4)."},"skills":["Opposite Midpoint and Three Squared Segments"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0082","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d5399727fa0a031b6f396664bce58bcba81fa884fbb1ad426ab4d583e0e4e8e","family_id":"qf-density13-triangle-median-equality","seed":202610137082,"semantic_template":"qf-density13-triangle-median-equality/[1,[\"1/2\",\"0/1\"],\"169/4\",\"197/4\"]"},"fields":{"title":"Opposite Midpoint and Three Squared Segments","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 7], [-6, 0], [7, 0]]. The points are distinct and noncollinear, vertex=1, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 0/1, 169/4, 169/4, 197/4)","explanation":"Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 169/4, 169/4, 197/4)."},"skills":["Opposite Midpoint and Three Squared Segments"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0083","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"998ecb49f53cca3bc84c180e09e54ece2a4f6d37fbcd02872b66841ee7e4ed41","family_id":"qf-density13-triangle-median-equality","seed":202610137083,"semantic_template":"qf-density13-triangle-median-equality/[2,[\"1/2\",\"0/1\"],\"225/4\",\"257/4\"]"},"fields":{"title":"Opposite Midpoint and Three Squared Segments","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[-7, 0], [0, 8], [8, 0]]. The points are distinct and noncollinear, vertex=2, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 0/1, 225/4, 225/4, 257/4)","explanation":"Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 225/4, 225/4, 257/4)."},"skills":["Opposite Midpoint and Three Squared Segments"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0084","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"36306c18005b6c9474f12fae31e1ffb81f027ed42e24d1d71dd5c8be4f9c4764","family_id":"qf-density13-triangle-median-equality","seed":202610137084,"semantic_template":"qf-density13-triangle-median-equality/[3,[\"1/2\",\"0/1\"],\"289/4\",\"325/4\"]"},"fields":{"title":"Opposite Midpoint and Three Squared Segments","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[-8, 0], [9, 0], [0, 9]]. The points are distinct and noncollinear, vertex=3, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 0/1, 289/4, 289/4, 325/4)","explanation":"Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 289/4, 289/4, 325/4)."},"skills":["Opposite Midpoint and Three Squared Segments"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0085","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96dd1718cb375aa1c205dfcb5c6570f7ca040a68ef3f96eb609b86bdd41b692b","family_id":"qf-density13-triangle-median-equality","seed":202610137085,"semantic_template":"qf-density13-triangle-median-equality/[1,[\"1/2\",\"0/1\"],\"361/4\",\"401/4\"]"},"fields":{"title":"Opposite Midpoint and Three Squared Segments","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 10], [-9, 0], [10, 0]]. The points are distinct and noncollinear, vertex=1, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 0/1, 361/4, 361/4, 401/4)","explanation":"Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 361/4, 361/4, 401/4)."},"skills":["Opposite Midpoint and Three Squared Segments"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0086","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d879c0155374a52e5fada7d7c366e8f5e8592c5e20e07e2521b7ffa4a5373c28","family_id":"qf-density13-triangle-median-equality","seed":202610137086,"semantic_template":"qf-density13-triangle-median-equality/[2,[\"1/2\",\"0/1\"],\"441/4\",\"485/4\"]"},"fields":{"title":"Opposite Midpoint and Three Squared Segments","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[-10, 0], [0, 11], [11, 0]]. The points are distinct and noncollinear, vertex=2, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 0/1, 441/4, 441/4, 485/4)","explanation":"Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 441/4, 441/4, 485/4)."},"skills":["Opposite Midpoint and Three Squared Segments"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0087","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"46ffc1565945c65f0ce30a1a29d83ddce53e7f3d3807f20e27cf1ce85294e924","family_id":"qf-density13-triangle-median-equality","seed":202610137087,"semantic_template":"qf-density13-triangle-median-equality/[3,[\"1/2\",\"0/1\"],\"529/4\",\"577/4\"]"},"fields":{"title":"Opposite Midpoint and Three Squared Segments","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[-11, 0], [12, 0], [0, 12]]. The points are distinct and noncollinear, vertex=3, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 0/1, 529/4, 529/4, 577/4)","explanation":"Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 529/4, 529/4, 577/4)."},"skills":["Opposite Midpoint and Three Squared Segments"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0088","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8db0b38c8ac3f2490dd54372354d2cf04cd48f5d61329e06ce589d40f2856400","family_id":"qf-density13-triangle-median-equality","seed":202610137088,"semantic_template":"qf-density13-triangle-median-equality/[1,[\"1/2\",\"0/1\"],\"625/4\",\"677/4\"]"},"fields":{"title":"Opposite Midpoint and Three Squared Segments","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 13], [-12, 0], [13, 0]]. The points are distinct and noncollinear, vertex=1, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 0/1, 625/4, 625/4, 677/4)","explanation":"Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 625/4, 625/4, 677/4)."},"skills":["Opposite Midpoint and Three Squared Segments"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0089","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2e7e1491ea3d43668d094a3f680e0d1247efe2193806e700536572810b59a9a","family_id":"qf-density13-triangle-median-equality","seed":202610137089,"semantic_template":"qf-density13-triangle-median-equality/[2,[\"1/2\",\"0/1\"],\"729/4\",\"785/4\"]"},"fields":{"title":"Opposite Midpoint and Three Squared Segments","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[-13, 0], [0, 14], [14, 0]]. The points are distinct and noncollinear, vertex=2, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 0/1, 729/4, 729/4, 785/4)","explanation":"Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 729/4, 729/4, 785/4)."},"skills":["Opposite Midpoint and Three Squared Segments"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0090","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"459cd13c8d0a09c7e6cf3c7d4948aa74fd70f8f883cc2f1b879485daf16b812b","family_id":"qf-density13-triangle-median-equality","seed":202610137090,"semantic_template":"qf-density13-triangle-median-equality/[3,[\"1/2\",\"0/1\"],\"841/4\",\"901/4\"]"},"fields":{"title":"Opposite Midpoint and Three Squared Segments","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[-14, 0], [15, 0], [0, 15]]. The points are distinct and noncollinear, vertex=3, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 0/1, 841/4, 841/4, 901/4)","explanation":"Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 841/4, 841/4, 901/4)."},"skills":["Opposite Midpoint and Three Squared Segments"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0091","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a94d648519aac674c90870d8c8119b8c57912c4796938f09e9acb1de672f564d","family_id":"qf-density13-triangle-median-equality","seed":202610137091,"semantic_template":"qf-density13-triangle-median-equality/[1,[\"1/2\",\"0/1\"],\"961/4\",\"1025/4\"]"},"fields":{"title":"Opposite Midpoint and Three Squared Segments","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 16], [-15, 0], [16, 0]]. The points are distinct and noncollinear, vertex=1, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 0/1, 961/4, 961/4, 1025/4)","explanation":"Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 961/4, 961/4, 1025/4)."},"skills":["Opposite Midpoint and Three Squared Segments"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0092","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5ab56d166f15aa49ef10817d1727e0fbf48d56e340cdbf1d050bf7288bd3089","family_id":"qf-density13-triangle-median-equality","seed":202610137092,"semantic_template":"qf-density13-triangle-median-equality/[2,[\"1/2\",\"0/1\"],\"1089/4\",\"1157/4\"]"},"fields":{"title":"Opposite Midpoint and Three Squared Segments","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[-16, 0], [0, 17], [17, 0]]. The points are distinct and noncollinear, vertex=2, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 0/1, 1089/4, 1089/4, 1157/4)","explanation":"Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 1089/4, 1089/4, 1157/4)."},"skills":["Opposite Midpoint and Three Squared Segments"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0093","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81592706d57282325273778b49e56a64e77cc01c12b632234a739706b486b533","family_id":"qf-density13-triangle-median-equality","seed":202610137093,"semantic_template":"qf-density13-triangle-median-equality/[3,[\"1/2\",\"0/1\"],\"1225/4\",\"1297/4\"]"},"fields":{"title":"Opposite Midpoint and Three Squared Segments","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[-17, 0], [18, 0], [0, 18]]. The points are distinct and noncollinear, vertex=3, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 0/1, 1225/4, 1225/4, 1297/4)","explanation":"Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 1225/4, 1225/4, 1297/4)."},"skills":["Opposite Midpoint and Three Squared Segments"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0094","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b87a075721a55d8a2ff802122a4d556542af1deae90db00eed9d528e11c8656","family_id":"qf-density13-triangle-median-equality","seed":202610137094,"semantic_template":"qf-density13-triangle-median-equality/[1,[\"1/2\",\"0/1\"],\"1369/4\",\"1445/4\"]"},"fields":{"title":"Opposite Midpoint and Three Squared Segments","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 19], [-18, 0], [19, 0]]. The points are distinct and noncollinear, vertex=1, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 0/1, 1369/4, 1369/4, 1445/4)","explanation":"Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 1369/4, 1369/4, 1445/4)."},"skills":["Opposite Midpoint and Three Squared Segments"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0095","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c4a2e0d9a1973b25ad04352931f68304b89f43350c9a289946377d8c0ddc4cb","family_id":"qf-density13-triangle-median-equality","seed":202610137095,"semantic_template":"qf-density13-triangle-median-equality/[2,[\"1/2\",\"0/1\"],\"1521/4\",\"1601/4\"]"},"fields":{"title":"Opposite Midpoint and Three Squared Segments","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[-19, 0], [0, 20], [20, 0]]. The points are distinct and noncollinear, vertex=2, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 0/1, 1521/4, 1521/4, 1601/4)","explanation":"Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 1521/4, 1521/4, 1601/4)."},"skills":["Opposite Midpoint and Three Squared Segments"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0096","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"500a6bece0e8fcb8cd8fcd52ecad23dc67ab5f264ff48aef9023c5930369b9a4","family_id":"qf-density13-triangle-median-equality","seed":202610137096,"semantic_template":"qf-density13-triangle-median-equality/[3,[\"1/2\",\"0/1\"],\"1681/4\",\"1765/4\"]"},"fields":{"title":"Opposite Midpoint and Three Squared Segments","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[-20, 0], [21, 0], [0, 21]]. The points are distinct and noncollinear, vertex=3, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 0/1, 1681/4, 1681/4, 1765/4)","explanation":"Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 1681/4, 1681/4, 1765/4)."},"skills":["Opposite Midpoint and Three Squared Segments"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0097","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7dbe8e299e442f45e8c63cdac644f07403069c0351e326ba4a304d58cfdcfa8","family_id":"qf-density13-triangle-median-equality","seed":202610137097,"semantic_template":"qf-density13-triangle-median-equality/[1,[\"1/2\",\"0/1\"],\"1849/4\",\"1937/4\"]"},"fields":{"title":"Opposite Midpoint and Three Squared Segments","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 22], [-21, 0], [22, 0]]. The points are distinct and noncollinear, vertex=1, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 0/1, 1849/4, 1849/4, 1937/4)","explanation":"Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 1849/4, 1849/4, 1937/4)."},"skills":["Opposite Midpoint and Three Squared Segments"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0098","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"807c9d3121bc2e3a80b3536956f1ee29b246762b2a07760bedbfb2bbe45f37b0","family_id":"qf-density13-triangle-median-equality","seed":202610137098,"semantic_template":"qf-density13-triangle-median-equality/[2,[\"1/2\",\"0/1\"],\"2025/4\",\"2117/4\"]"},"fields":{"title":"Opposite Midpoint and Three Squared Segments","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[-22, 0], [0, 23], [23, 0]]. The points are distinct and noncollinear, vertex=2, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 0/1, 2025/4, 2025/4, 2117/4)","explanation":"Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 2025/4, 2025/4, 2117/4)."},"skills":["Opposite Midpoint and Three Squared Segments"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0099","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d403407f3cbd85767478db9415a9f35f4b57af67dedb56b8a32d15f54bcf3972","family_id":"qf-density13-triangle-median-equality","seed":202610137099,"semantic_template":"qf-density13-triangle-median-equality/[3,[\"1/2\",\"0/1\"],\"2209/4\",\"2305/4\"]"},"fields":{"title":"Opposite Midpoint and Three Squared Segments","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[-23, 0], [24, 0], [0, 24]]. The points are distinct and noncollinear, vertex=3, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 0/1, 2209/4, 2209/4, 2305/4)","explanation":"Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 2209/4, 2209/4, 2305/4)."},"skills":["Opposite Midpoint and Three Squared Segments"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r07-triangle-0100","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a15491399506a0c645d8b1945358a65bc368ed751e1bef923c98f0558c15def4","family_id":"qf-density13-triangle-median-equality","seed":202610137100,"semantic_template":"qf-density13-triangle-median-equality/[1,[\"1/2\",\"0/1\"],\"2401/4\",\"2501/4\"]"},"fields":{"title":"Opposite Midpoint and Three Squared Segments","stem":"In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 25], [-24, 0], [25, 0]]. The points are distinct and noncollinear, vertex=1, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 0/1, 2401/4, 2401/4, 2501/4)","explanation":"Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 2401/4, 2401/4, 2501/4)."},"skills":["Opposite Midpoint and Three Squared Segments"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}