{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261009-advance-20261013-r07-triangle-0032", "pilot_number": "R0012-06582", "number": 6582, "family_id": "qf-density13-triangle-opposite-side-squares", "title": { "zh-Hant": "第 6582 題：標記頂點的對邊端點與平方", "en": "Question 6582: Labeled Opposite Endpoints and Squared Side" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261009-advance-20261013-r07-triangle-0032", "batch_id": "batch-20261009-advance-20261013-r07-triangle-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-triangle" ], "title_zh": "標記頂點的對邊端點與平方", "stem_zh": "歐氏等單位正交座標；A、B、C按頂點索引1、2、3排列，points=[[0, 1], [0, 0], [7, 0]]。已要求三點相異且非共線，vertex=1。依序列 (另外兩個頂點的索引, 對邊L2平方距離)，不要將平方當距離。", "answer": { "kind": "expression", "value": "([2, 3], 49)", "explanation_zh": "對邊端點與平方距離=([2, 3], 49)。依固定標記取另外兩點，再加座標差平方；有限座標檢查非一般幾何證明，不從示意圖讀量值。" }, "difficulty": { "band": "upper_secondary", "level": 1, "skills": [ "標記頂點的對邊端點與平方" ] }, "variant": { "family_id": "qf-density13-triangle-opposite-side-squares", "seed": 202610137032, "parameters": { "points": [ [ 0, 1 ], [ 0, 0 ], [ 7, 0 ] ], "vertex": 1, "frame": "orthogonal_equal_unit", "operation": "density13_triangle_opposite_side_squares", "domain": "finite_labelled_euclidean_triangle", "semantic_template": "qf-density13-triangle-opposite-side-squares/[1,\"49\"]" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_integer_rational_arithmetic", "producer_finite_enumeration", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-09", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "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." } }, "givens": [ { "id": "G001", "text": { "zh-Hant": "歐氏等單位正交座標；A、B、C按頂點索引1、2、3排列，points=[[0, 1], [0, 0], [7, 0]]。已要求三點相異且非共線，vertex=1。依序列 (另外兩個頂點的索引, 對邊L2平方距離)，不要將平方當距離。", "en": "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." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "[[0, 1], [0, 0], [7, 0]]", "after": "original vertex labels1,2,3 in an equal-unit orthogonal Euclidean frame", "relation": "implies", "rule_id": "labelled_triangle_coordinate_scope", "conditions": [], "explanation": { "zh-Hant": "保留A、B、C原順序，等單位且正交才使用此座標平方公式；不由示意圖讀數，候選頂點未必是合格三角形。", "en": "Retain the original A,B,C order;the squared-coordinate formula uses equal units and orthogonal axes.Do not read values from a sketch or assume candidate vertices already form a triangle." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "[[0, 1], [0, 0], [7, 0]]", "after": "three distinct points=是", "relation": "implies", "rule_id": "distinct_triangle_vertex_check", "conditions": [], "explanation": { "zh-Hant": "相異資格與非共線資格分開；三點共線可以彼此相異，重複頂點則不合非退化三角形。", "en": "Check distinctness separately from noncollinearity;collinear points may be distinct,while repeated vertices do not form a proper triangle." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "ABC first vector component0 := sub(0, 0)", "after": "ABC first vector component0 = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "第一基向量是第二頂點減第一頂點。", "en": "The first edge vector is the second vertex minus the first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 ABC first vector component0，不以小數近似替代。", "en": "Recompute ABC first vector component0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "0", "0" ], "output": "0", "label": "ABC first vector component0" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "ABC first vector component1 := sub(0, 1)", "after": "ABC first vector component1 = -1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "第一基向量是第二頂點減第一頂點。", "en": "The first edge vector is the second vertex minus the first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 ABC first vector component1，不以小數近似替代。", "en": "Recompute ABC first vector component1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "0", "1" ], "output": "-1", "label": "ABC first vector component1" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "ABC second vector component0 := sub(7, 0)", "after": "ABC second vector component0 = 7", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "第二基向量是第三頂點減同一第一頂點。", "en": "The second edge vector is the third vertex minus that same first vertex." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 ABC second vector component0，不以小數近似替代。", "en": "Recompute ABC second vector component0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "7", "0" ], "output": "7", "label": "ABC second vector component0" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "ABC second vector component1 := sub(0, 1)", "after": "ABC second vector component1 = -1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "第二基向量是第三頂點減同一第一頂點。", "en": "The second edge vector is the third vertex minus that same first vertex." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 ABC second vector component1，不以小數近似替代。", "en": "Recompute ABC second vector component1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "0", "1" ], "output": "-1", "label": "ABC second vector component1" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "ABC cross positive term := mul(0, -1)", "after": "ABC cross positive term = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先算u_x v_y。", "en": "Compute u_x v_y first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 ABC cross positive term，不以小數近似替代。", "en": "Recompute ABC cross positive term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "-1" ], "output": "0", "label": "ABC cross positive term" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "ABC cross negative term := mul(-1, 7)", "after": "ABC cross negative term = -7", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "另算u_y v_x，減號留到下一步。", "en": "Compute u_y v_x separately,retaining subtraction for the next step." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 ABC cross negative term，不以小數近似替代。", "en": "Recompute ABC cross negative term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-1", "7" ], "output": "-7", "label": "ABC cross negative term" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "ABC signed determinant := sub(0, -7)", "after": "ABC signed determinant = 7", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "相減給有號二倍面積，不能取絕對值抹去標記方向。", "en": "Subtract to obtain signed double area;do not erase label orientation with an absolute value." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 ABC signed determinant，不以小數近似替代。", "en": "Recompute ABC signed determinant with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "0", "-7" ], "output": "7", "label": "ABC signed determinant" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "distinct=是;signed determinant=7", "after": "proper triangle=是", "relation": "implies", "rule_id": "triangle_nondegenerate_coordinate_qualification", "conditions": [], "explanation": { "zh-Hant": "非退化要求相異且行列式非零；只對這組明示座標給判定，不冒稱一般幾何構造證明。", "en": "A proper triangle needs distinct vertices and nonzero determinant;this judges the supplied coordinates,not a general geometric construction proof." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "marked vertex=1", "after": "opposite endpoints=[2, 3]", "relation": "implies", "rule_id": "original_label_opposite_side", "conditions": [], "explanation": { "zh-Hant": "對邊端點是另外兩個原標記，不依座標或邊長重新排序。", "en": "The opposite-side endpoints are the other two original labels,not a reordered coordinate or side-length list." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "opposite side coordinate difference0 := sub(0, 7)", "after": "opposite side coordinate difference0 = -7", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先取同分量座標差，允許中點有理數。", "en": "Subtract matching coordinates,allowing rational midpoint coordinates." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 opposite side coordinate difference0，不以小數近似替代。", "en": "Recompute opposite side coordinate difference0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "0", "7" ], "output": "-7", "label": "opposite side coordinate difference0" } }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "opposite side difference squared0 := square(-7)", "after": "opposite side difference squared0 = 49", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "逐分量平方，差的正負不影響平方長度。", "en": "Square each difference;its sign does not affect squared length." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 opposite side difference squared0，不以小數近似替代。", "en": "Recompute opposite side difference squared0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "square", "inputs": [ "-7" ], "output": "49", "label": "opposite side difference squared0" } }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "opposite side coordinate difference1 := sub(0, 0)", "after": "opposite side coordinate difference1 = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先取同分量座標差，允許中點有理數。", "en": "Subtract matching coordinates,allowing rational midpoint coordinates." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 opposite side coordinate difference1，不以小數近似替代。", "en": "Recompute opposite side coordinate difference1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "0", "0" ], "output": "0", "label": "opposite side coordinate difference1" } }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "opposite side difference squared1 := square(0)", "after": "opposite side difference squared1 = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "逐分量平方，差的正負不影響平方長度。", "en": "Square each difference;its sign does not affect squared length." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 opposite side difference squared1，不以小數近似替代。", "en": "Recompute opposite side difference squared1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "square", "inputs": [ "0" ], "output": "0", "label": "opposite side difference squared1" } }, { "id": "S016", "uses": [ "G001", "S015" ], "before": "opposite side squared distance := sum(49, 0)", "after": "opposite side squared distance = 49", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依等單位正交L2關係相加兩平方。", "en": "Add the two squares in the equal-unit orthogonal L2 frame." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 opposite side squared distance，不以小數近似替代。", "en": "Recompute opposite side squared distance with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "49", "0" ], "output": "49", "label": "opposite side squared distance" } }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "([2, 3], 49)", "after": "second field is squared length,not length", "relation": "implies", "rule_id": "opposite_side_square_not_length", "conditions": [], "explanation": { "zh-Hant": "不把平方距離直接稱邊長，也不因頂點選擇改變端點標記次序。", "en": "Do not call squared distance the side length or reorder endpoint labels because of the chosen vertex." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null } ], "math_model": { "result": [ [ 2, 3 ], "49" ] }, "conclusion": { "canonical_value": "([2, 3], 49)", "explanation": { "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。", "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged." } }, "definition_note": { "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。", "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate." }, "source": { "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db", "question_binding": { "batch_id": "batch-20261009-advance-20261013-r07-triangle-foundations", "batch_path": "public/data/questions/batches/batch-20261009-advance-20261013-r07-triangle-foundations/questions.jsonl", "batch_sha256": "5063b1fb565b0934cadf3e54de8bfbf6e9cff8d3ef786ecc721a592ca4db0578", "original_line": 32, "original_line_sha256": "b48fe4f3533cba66e50dd34d7a7be50a493a42732ccb6ee2b20455f93bb301d3", "question_compact_sha256": "79ff998847062aa58b88953136010b4e59ead61b7ce265eb1cf289efb26c19da", "english_row_compact_sha256": "d72a767a0789c32692a594979de9e9aa751648976e3f04f6fdeac722975fc219", "english_manifest_path": "public/data/i18n/editions/batch-20261009-advance-20261013-r07-triangle-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261009-advance-20261013-r07-triangle-foundations.en.jsonl" }, "mko_bindings": { "mko-triangle": { "sha256": "1556dabd42bf5d132f41296684eb2b037900a511e659c48627bee6d0c4dcd461", "english_sha256": "e4a28d8e7440ec3005d4c989132d0cfa5a0c60bf2440ebcd24ca5c2e886f4b5f", "original_path": "public/data/mko/triangle.json", "english_path": "public/data/i18n/editions/mko-triangle.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "c5c612ca91797c328f4489939ec6f7d885385e63ea21b334c09c421f17933d99", "original_candidate_sha256": "e864062df63bc09e8fba39c15126e93ad5b37274af209c9324139cbb87a6302b", "original_candidate_line_sha256": "e864062df63bc09e8fba39c15126e93ad5b37274af209c9324139cbb87a6302b" }, "historical_source_state": { "review_state": { "derivation": "candidate_authored_and_locally_checked", "source_question_state": "candidate", "independent_AI_or_human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN" }, "scope_note": { "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。", "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged." }, "label": "Frozen producer metadata; current per-question review is separate below." }, "revision": { "id": "review-20261026-005-p13", "review_date": "2026-10-26", "owner_authorized_reader_publication": true, "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.", "review_receipts": { "semantic_review_sha256": "90dbc494b2d34ef51f412dcff63cfcc1a2215d038bff71f38f5dded4c64faf00", "question_first_witness_sha256": "b72a3b8bcf10f584396e0dd5198c657b0934a71864e5a3fa6321d7a613aef9f9" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 17, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "adb2abbc93991ebffc18905839a9849777257c01760ab4c5f01136b61a984f1d", "arithmetic_steps": 12 }, "uses_answer_as_premise": false, "human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN", "Canonical_truth_promotion": false, "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged." } }