{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0001","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"候選頂點的相異與共線資格","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, 0], [0, 0], [1, 1]]。此為候選頂點資料。依序答 (三點是否相異, 有號(B−A)×(C−A), 相異且非共線是否)，允許退化候選。","answer":{"kind":"expression","value":"(否, 0, 否)","explanation_zh":"頂點資格與共線見證=(否, 0, 否)。用精確座標及行列式分開核相異與非共線；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":["候選頂點的相異與共線資格"]},"variant":{"family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137001,"parameters":{"points":[[0,0],[0,0],[1,1]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_coordinate_qualification","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-coordinate-qualification/[false,\"0\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0002","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"候選頂點的相異與共線資格","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, 0], [1, 1], [2, 2]]。此為候選頂點資料。依序答 (三點是否相異, 有號(B−A)×(C−A), 相異且非共線是否)，允許退化候選。","answer":{"kind":"expression","value":"(是, 0, 否)","explanation_zh":"頂點資格與共線見證=(是, 0, 否)。用精確座標及行列式分開核相異與非共線；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":["候選頂點的相異與共線資格"]},"variant":{"family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137002,"parameters":{"points":[[0,0],[1,1],[2,2]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_coordinate_qualification","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"0\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0003","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"候選頂點的相異與共線資格","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, 0], [1, 0], [0, -12]]。此為候選頂點資料。依序答 (三點是否相異, 有號(B−A)×(C−A), 相異且非共線是否)，允許退化候選。","answer":{"kind":"expression","value":"(是, -12, 是)","explanation_zh":"頂點資格與共線見證=(是, -12, 是)。用精確座標及行列式分開核相異與非共線；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":["候選頂點的相異與共線資格"]},"variant":{"family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137003,"parameters":{"points":[[0,0],[1,0],[0,-12]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_coordinate_qualification","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"-12\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0004","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"候選頂點的相異與共線資格","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, 0], [1, 0], [0, -11]]。此為候選頂點資料。依序答 (三點是否相異, 有號(B−A)×(C−A), 相異且非共線是否)，允許退化候選。","answer":{"kind":"expression","value":"(是, -11, 是)","explanation_zh":"頂點資格與共線見證=(是, -11, 是)。用精確座標及行列式分開核相異與非共線；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":["候選頂點的相異與共線資格"]},"variant":{"family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137004,"parameters":{"points":[[0,0],[1,0],[0,-11]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_coordinate_qualification","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"-11\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0005","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"候選頂點的相異與共線資格","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, 0], [1, 0], [0, -10]]。此為候選頂點資料。依序答 (三點是否相異, 有號(B−A)×(C−A), 相異且非共線是否)，允許退化候選。","answer":{"kind":"expression","value":"(是, -10, 是)","explanation_zh":"頂點資格與共線見證=(是, -10, 是)。用精確座標及行列式分開核相異與非共線；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":["候選頂點的相異與共線資格"]},"variant":{"family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137005,"parameters":{"points":[[0,0],[1,0],[0,-10]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_coordinate_qualification","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"-10\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0006","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"候選頂點的相異與共線資格","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, 0], [1, 0], [0, -9]]。此為候選頂點資料。依序答 (三點是否相異, 有號(B−A)×(C−A), 相異且非共線是否)，允許退化候選。","answer":{"kind":"expression","value":"(是, -9, 是)","explanation_zh":"頂點資格與共線見證=(是, -9, 是)。用精確座標及行列式分開核相異與非共線；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":["候選頂點的相異與共線資格"]},"variant":{"family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137006,"parameters":{"points":[[0,0],[1,0],[0,-9]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_coordinate_qualification","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"-9\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0007","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"候選頂點的相異與共線資格","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, 0], [1, 0], [0, -8]]。此為候選頂點資料。依序答 (三點是否相異, 有號(B−A)×(C−A), 相異且非共線是否)，允許退化候選。","answer":{"kind":"expression","value":"(是, -8, 是)","explanation_zh":"頂點資格與共線見證=(是, -8, 是)。用精確座標及行列式分開核相異與非共線；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":["候選頂點的相異與共線資格"]},"variant":{"family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137007,"parameters":{"points":[[0,0],[1,0],[0,-8]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_coordinate_qualification","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"-8\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0008","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"候選頂點的相異與共線資格","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, 0], [1, 0], [0, -7]]。此為候選頂點資料。依序答 (三點是否相異, 有號(B−A)×(C−A), 相異且非共線是否)，允許退化候選。","answer":{"kind":"expression","value":"(是, -7, 是)","explanation_zh":"頂點資格與共線見證=(是, -7, 是)。用精確座標及行列式分開核相異與非共線；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":["候選頂點的相異與共線資格"]},"variant":{"family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137008,"parameters":{"points":[[0,0],[1,0],[0,-7]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_coordinate_qualification","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"-7\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0009","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"候選頂點的相異與共線資格","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, 0], [1, 0], [0, -6]]。此為候選頂點資料。依序答 (三點是否相異, 有號(B−A)×(C−A), 相異且非共線是否)，允許退化候選。","answer":{"kind":"expression","value":"(是, -6, 是)","explanation_zh":"頂點資格與共線見證=(是, -6, 是)。用精確座標及行列式分開核相異與非共線；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":2,"skills":["候選頂點的相異與共線資格"]},"variant":{"family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137009,"parameters":{"points":[[0,0],[1,0],[0,-6]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_coordinate_qualification","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"-6\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0010","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"候選頂點的相異與共線資格","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, 0], [1, 0], [0, -5]]。此為候選頂點資料。依序答 (三點是否相異, 有號(B−A)×(C−A), 相異且非共線是否)，允許退化候選。","answer":{"kind":"expression","value":"(是, -5, 是)","explanation_zh":"頂點資格與共線見證=(是, -5, 是)。用精確座標及行列式分開核相異與非共線；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":2,"skills":["候選頂點的相異與共線資格"]},"variant":{"family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137010,"parameters":{"points":[[0,0],[1,0],[0,-5]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_coordinate_qualification","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"-5\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0011","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"候選頂點的相異與共線資格","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, 0], [1, 0], [0, -4]]。此為候選頂點資料。依序答 (三點是否相異, 有號(B−A)×(C−A), 相異且非共線是否)，允許退化候選。","answer":{"kind":"expression","value":"(是, -4, 是)","explanation_zh":"頂點資格與共線見證=(是, -4, 是)。用精確座標及行列式分開核相異與非共線；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":2,"skills":["候選頂點的相異與共線資格"]},"variant":{"family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137011,"parameters":{"points":[[0,0],[1,0],[0,-4]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_coordinate_qualification","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"-4\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0012","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"候選頂點的相異與共線資格","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, 0], [1, 0], [0, -3]]。此為候選頂點資料。依序答 (三點是否相異, 有號(B−A)×(C−A), 相異且非共線是否)，允許退化候選。","answer":{"kind":"expression","value":"(是, -3, 是)","explanation_zh":"頂點資格與共線見證=(是, -3, 是)。用精確座標及行列式分開核相異與非共線；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":2,"skills":["候選頂點的相異與共線資格"]},"variant":{"family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137012,"parameters":{"points":[[0,0],[1,0],[0,-3]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_coordinate_qualification","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"-3\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0013","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"候選頂點的相異與共線資格","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, 0], [1, 0], [0, -2]]。此為候選頂點資料。依序答 (三點是否相異, 有號(B−A)×(C−A), 相異且非共線是否)，允許退化候選。","answer":{"kind":"expression","value":"(是, -2, 是)","explanation_zh":"頂點資格與共線見證=(是, -2, 是)。用精確座標及行列式分開核相異與非共線；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":2,"skills":["候選頂點的相異與共線資格"]},"variant":{"family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137013,"parameters":{"points":[[0,0],[1,0],[0,-2]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_coordinate_qualification","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"-2\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0014","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"候選頂點的相異與共線資格","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, 0], [1, 0], [0, -1]]。此為候選頂點資料。依序答 (三點是否相異, 有號(B−A)×(C−A), 相異且非共線是否)，允許退化候選。","answer":{"kind":"expression","value":"(是, -1, 是)","explanation_zh":"頂點資格與共線見證=(是, -1, 是)。用精確座標及行列式分開核相異與非共線；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":2,"skills":["候選頂點的相異與共線資格"]},"variant":{"family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137014,"parameters":{"points":[[0,0],[1,0],[0,-1]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_coordinate_qualification","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"-1\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0015","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"候選頂點的相異與共線資格","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, 0], [1, 0], [0, 1]]。此為候選頂點資料。依序答 (三點是否相異, 有號(B−A)×(C−A), 相異且非共線是否)，允許退化候選。","answer":{"kind":"expression","value":"(是, 1, 是)","explanation_zh":"頂點資格與共線見證=(是, 1, 是)。用精確座標及行列式分開核相異與非共線；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":2,"skills":["候選頂點的相異與共線資格"]},"variant":{"family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137015,"parameters":{"points":[[0,0],[1,0],[0,1]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_coordinate_qualification","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"1\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0016","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"候選頂點的相異與共線資格","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, 0], [1, 0], [0, 2]]。此為候選頂點資料。依序答 (三點是否相異, 有號(B−A)×(C−A), 相異且非共線是否)，允許退化候選。","answer":{"kind":"expression","value":"(是, 2, 是)","explanation_zh":"頂點資格與共線見證=(是, 2, 是)。用精確座標及行列式分開核相異與非共線；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":2,"skills":["候選頂點的相異與共線資格"]},"variant":{"family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137016,"parameters":{"points":[[0,0],[1,0],[0,2]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_coordinate_qualification","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"2\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0017","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"候選頂點的相異與共線資格","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, 0], [1, 0], [0, 3]]。此為候選頂點資料。依序答 (三點是否相異, 有號(B−A)×(C−A), 相異且非共線是否)，允許退化候選。","answer":{"kind":"expression","value":"(是, 3, 是)","explanation_zh":"頂點資格與共線見證=(是, 3, 是)。用精確座標及行列式分開核相異與非共線；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":2,"skills":["候選頂點的相異與共線資格"]},"variant":{"family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137017,"parameters":{"points":[[0,0],[1,0],[0,3]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_coordinate_qualification","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"3\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0018","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"候選頂點的相異與共線資格","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, 0], [1, 0], [0, 4]]。此為候選頂點資料。依序答 (三點是否相異, 有號(B−A)×(C−A), 相異且非共線是否)，允許退化候選。","answer":{"kind":"expression","value":"(是, 4, 是)","explanation_zh":"頂點資格與共線見證=(是, 4, 是)。用精確座標及行列式分開核相異與非共線；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":3,"skills":["候選頂點的相異與共線資格"]},"variant":{"family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137018,"parameters":{"points":[[0,0],[1,0],[0,4]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_coordinate_qualification","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"4\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0019","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"候選頂點的相異與共線資格","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, 0], [1, 0], [0, 5]]。此為候選頂點資料。依序答 (三點是否相異, 有號(B−A)×(C−A), 相異且非共線是否)，允許退化候選。","answer":{"kind":"expression","value":"(是, 5, 是)","explanation_zh":"頂點資格與共線見證=(是, 5, 是)。用精確座標及行列式分開核相異與非共線；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":3,"skills":["候選頂點的相異與共線資格"]},"variant":{"family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137019,"parameters":{"points":[[0,0],[1,0],[0,5]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_coordinate_qualification","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"5\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0020","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"候選頂點的相異與共線資格","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, 0], [1, 0], [0, 6]]。此為候選頂點資料。依序答 (三點是否相異, 有號(B−A)×(C−A), 相異且非共線是否)，允許退化候選。","answer":{"kind":"expression","value":"(是, 6, 是)","explanation_zh":"頂點資格與共線見證=(是, 6, 是)。用精確座標及行列式分開核相異與非共線；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":3,"skills":["候選頂點的相異與共線資格"]},"variant":{"family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137020,"parameters":{"points":[[0,0],[1,0],[0,6]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_coordinate_qualification","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"6\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0021","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"候選頂點的相異與共線資格","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, 0], [1, 0], [0, 7]]。此為候選頂點資料。依序答 (三點是否相異, 有號(B−A)×(C−A), 相異且非共線是否)，允許退化候選。","answer":{"kind":"expression","value":"(是, 7, 是)","explanation_zh":"頂點資格與共線見證=(是, 7, 是)。用精確座標及行列式分開核相異與非共線；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":3,"skills":["候選頂點的相異與共線資格"]},"variant":{"family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137021,"parameters":{"points":[[0,0],[1,0],[0,7]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_coordinate_qualification","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"7\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0022","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"候選頂點的相異與共線資格","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, 0], [1, 0], [0, 8]]。此為候選頂點資料。依序答 (三點是否相異, 有號(B−A)×(C−A), 相異且非共線是否)，允許退化候選。","answer":{"kind":"expression","value":"(是, 8, 是)","explanation_zh":"頂點資格與共線見證=(是, 8, 是)。用精確座標及行列式分開核相異與非共線；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":3,"skills":["候選頂點的相異與共線資格"]},"variant":{"family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137022,"parameters":{"points":[[0,0],[1,0],[0,8]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_coordinate_qualification","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"8\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0023","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"候選頂點的相異與共線資格","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, 0], [1, 0], [0, 9]]。此為候選頂點資料。依序答 (三點是否相異, 有號(B−A)×(C−A), 相異且非共線是否)，允許退化候選。","answer":{"kind":"expression","value":"(是, 9, 是)","explanation_zh":"頂點資格與共線見證=(是, 9, 是)。用精確座標及行列式分開核相異與非共線；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":3,"skills":["候選頂點的相異與共線資格"]},"variant":{"family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137023,"parameters":{"points":[[0,0],[1,0],[0,9]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_coordinate_qualification","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"9\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0024","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"候選頂點的相異與共線資格","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, 0], [1, 0], [0, 10]]。此為候選頂點資料。依序答 (三點是否相異, 有號(B−A)×(C−A), 相異且非共線是否)，允許退化候選。","answer":{"kind":"expression","value":"(是, 10, 是)","explanation_zh":"頂點資格與共線見證=(是, 10, 是)。用精確座標及行列式分開核相異與非共線；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":3,"skills":["候選頂點的相異與共線資格"]},"variant":{"family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137024,"parameters":{"points":[[0,0],[1,0],[0,10]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_coordinate_qualification","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"10\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0025","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"候選頂點的相異與共線資格","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, 0], [1, 0], [0, 11]]。此為候選頂點資料。依序答 (三點是否相異, 有號(B−A)×(C−A), 相異且非共線是否)，允許退化候選。","answer":{"kind":"expression","value":"(是, 11, 是)","explanation_zh":"頂點資格與共線見證=(是, 11, 是)。用精確座標及行列式分開核相異與非共線；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":3,"skills":["候選頂點的相異與共線資格"]},"variant":{"family_id":"qf-density13-triangle-coordinate-qualification","seed":202610137025,"parameters":{"points":[[0,0],[1,0],[0,11]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_coordinate_qualification","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-coordinate-qualification/[true,\"11\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0026","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], [1, 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], 1)","explanation_zh":"對邊端點與平方距離=([2, 3], 1)。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":202610137026,"parameters":{"points":[[0,1],[0,0],[1,0]],"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,\"1\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0027","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, 0], [0, 1], [2, 0]]。已要求三點相異且非共線，vertex=2。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":"([1, 3], 4)","explanation_zh":"對邊端點與平方距離=([1, 3], 4)。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":202610137027,"parameters":{"points":[[0,0],[0,1],[2,0]],"vertex":2,"frame":"orthogonal_equal_unit","operation":"density13_triangle_opposite_side_squares","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-opposite-side-squares/[2,\"4\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0028","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, 0], [3, 0], [0, 1]]。已要求三點相異且非共線，vertex=3。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":"([1, 2], 9)","explanation_zh":"對邊端點與平方距離=([1, 2], 9)。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":202610137028,"parameters":{"points":[[0,0],[3,0],[0,1]],"vertex":3,"frame":"orthogonal_equal_unit","operation":"density13_triangle_opposite_side_squares","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-opposite-side-squares/[3,\"9\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0029","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], [4, 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], 16)","explanation_zh":"對邊端點與平方距離=([2, 3], 16)。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":202610137029,"parameters":{"points":[[0,1],[0,0],[4,0]],"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,\"16\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0030","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, 0], [0, 1], [5, 0]]。已要求三點相異且非共線，vertex=2。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":"([1, 3], 25)","explanation_zh":"對邊端點與平方距離=([1, 3], 25)。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":202610137030,"parameters":{"points":[[0,0],[0,1],[5,0]],"vertex":2,"frame":"orthogonal_equal_unit","operation":"density13_triangle_opposite_side_squares","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-opposite-side-squares/[2,\"25\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0031","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, 0], [6, 0], [0, 1]]。已要求三點相異且非共線，vertex=3。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":"([1, 2], 36)","explanation_zh":"對邊端點與平方距離=([1, 2], 36)。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":202610137031,"parameters":{"points":[[0,0],[6,0],[0,1]],"vertex":3,"frame":"orthogonal_equal_unit","operation":"density13_triangle_opposite_side_squares","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-opposite-side-squares/[3,\"36\"]"}},"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"}}
{"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0033","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, 0], [0, 1], [8, 0]]。已要求三點相異且非共線，vertex=2。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":"([1, 3], 64)","explanation_zh":"對邊端點與平方距離=([1, 3], 64)。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":202610137033,"parameters":{"points":[[0,0],[0,1],[8,0]],"vertex":2,"frame":"orthogonal_equal_unit","operation":"density13_triangle_opposite_side_squares","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-opposite-side-squares/[2,\"64\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0034","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, 0], [9, 0], [0, 1]]。已要求三點相異且非共線，vertex=3。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":"([1, 2], 81)","explanation_zh":"對邊端點與平方距離=([1, 2], 81)。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":2,"skills":["標記頂點的對邊端點與平方"]},"variant":{"family_id":"qf-density13-triangle-opposite-side-squares","seed":202610137034,"parameters":{"points":[[0,0],[9,0],[0,1]],"vertex":3,"frame":"orthogonal_equal_unit","operation":"density13_triangle_opposite_side_squares","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-opposite-side-squares/[3,\"81\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0035","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], [10, 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], 100)","explanation_zh":"對邊端點與平方距離=([2, 3], 100)。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":2,"skills":["標記頂點的對邊端點與平方"]},"variant":{"family_id":"qf-density13-triangle-opposite-side-squares","seed":202610137035,"parameters":{"points":[[0,1],[0,0],[10,0]],"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,\"100\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0036","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, 0], [0, 1], [11, 0]]。已要求三點相異且非共線，vertex=2。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":"([1, 3], 121)","explanation_zh":"對邊端點與平方距離=([1, 3], 121)。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":2,"skills":["標記頂點的對邊端點與平方"]},"variant":{"family_id":"qf-density13-triangle-opposite-side-squares","seed":202610137036,"parameters":{"points":[[0,0],[0,1],[11,0]],"vertex":2,"frame":"orthogonal_equal_unit","operation":"density13_triangle_opposite_side_squares","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-opposite-side-squares/[2,\"121\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0037","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, 0], [12, 0], [0, 1]]。已要求三點相異且非共線，vertex=3。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":"([1, 2], 144)","explanation_zh":"對邊端點與平方距離=([1, 2], 144)。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":2,"skills":["標記頂點的對邊端點與平方"]},"variant":{"family_id":"qf-density13-triangle-opposite-side-squares","seed":202610137037,"parameters":{"points":[[0,0],[12,0],[0,1]],"vertex":3,"frame":"orthogonal_equal_unit","operation":"density13_triangle_opposite_side_squares","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-opposite-side-squares/[3,\"144\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0038","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], [13, 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], 169)","explanation_zh":"對邊端點與平方距離=([2, 3], 169)。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":2,"skills":["標記頂點的對邊端點與平方"]},"variant":{"family_id":"qf-density13-triangle-opposite-side-squares","seed":202610137038,"parameters":{"points":[[0,1],[0,0],[13,0]],"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,\"169\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0039","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, 0], [0, 1], [14, 0]]。已要求三點相異且非共線，vertex=2。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":"([1, 3], 196)","explanation_zh":"對邊端點與平方距離=([1, 3], 196)。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":2,"skills":["標記頂點的對邊端點與平方"]},"variant":{"family_id":"qf-density13-triangle-opposite-side-squares","seed":202610137039,"parameters":{"points":[[0,0],[0,1],[14,0]],"vertex":2,"frame":"orthogonal_equal_unit","operation":"density13_triangle_opposite_side_squares","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-opposite-side-squares/[2,\"196\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0040","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, 0], [15, 0], [0, 1]]。已要求三點相異且非共線，vertex=3。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":"([1, 2], 225)","explanation_zh":"對邊端點與平方距離=([1, 2], 225)。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":2,"skills":["標記頂點的對邊端點與平方"]},"variant":{"family_id":"qf-density13-triangle-opposite-side-squares","seed":202610137040,"parameters":{"points":[[0,0],[15,0],[0,1]],"vertex":3,"frame":"orthogonal_equal_unit","operation":"density13_triangle_opposite_side_squares","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-opposite-side-squares/[3,\"225\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0041","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], [16, 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], 256)","explanation_zh":"對邊端點與平方距離=([2, 3], 256)。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":2,"skills":["標記頂點的對邊端點與平方"]},"variant":{"family_id":"qf-density13-triangle-opposite-side-squares","seed":202610137041,"parameters":{"points":[[0,1],[0,0],[16,0]],"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,\"256\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0042","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, 0], [0, 1], [17, 0]]。已要求三點相異且非共線，vertex=2。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":"([1, 3], 289)","explanation_zh":"對邊端點與平方距離=([1, 3], 289)。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":2,"skills":["標記頂點的對邊端點與平方"]},"variant":{"family_id":"qf-density13-triangle-opposite-side-squares","seed":202610137042,"parameters":{"points":[[0,0],[0,1],[17,0]],"vertex":2,"frame":"orthogonal_equal_unit","operation":"density13_triangle_opposite_side_squares","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-opposite-side-squares/[2,\"289\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0043","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, 0], [18, 0], [0, 1]]。已要求三點相異且非共線，vertex=3。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":"([1, 2], 324)","explanation_zh":"對邊端點與平方距離=([1, 2], 324)。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":3,"skills":["標記頂點的對邊端點與平方"]},"variant":{"family_id":"qf-density13-triangle-opposite-side-squares","seed":202610137043,"parameters":{"points":[[0,0],[18,0],[0,1]],"vertex":3,"frame":"orthogonal_equal_unit","operation":"density13_triangle_opposite_side_squares","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-opposite-side-squares/[3,\"324\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0044","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], [19, 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], 361)","explanation_zh":"對邊端點與平方距離=([2, 3], 361)。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":3,"skills":["標記頂點的對邊端點與平方"]},"variant":{"family_id":"qf-density13-triangle-opposite-side-squares","seed":202610137044,"parameters":{"points":[[0,1],[0,0],[19,0]],"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,\"361\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0045","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, 0], [0, 1], [20, 0]]。已要求三點相異且非共線，vertex=2。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":"([1, 3], 400)","explanation_zh":"對邊端點與平方距離=([1, 3], 400)。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":3,"skills":["標記頂點的對邊端點與平方"]},"variant":{"family_id":"qf-density13-triangle-opposite-side-squares","seed":202610137045,"parameters":{"points":[[0,0],[0,1],[20,0]],"vertex":2,"frame":"orthogonal_equal_unit","operation":"density13_triangle_opposite_side_squares","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-opposite-side-squares/[2,\"400\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0046","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, 0], [21, 0], [0, 1]]。已要求三點相異且非共線，vertex=3。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":"([1, 2], 441)","explanation_zh":"對邊端點與平方距離=([1, 2], 441)。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":3,"skills":["標記頂點的對邊端點與平方"]},"variant":{"family_id":"qf-density13-triangle-opposite-side-squares","seed":202610137046,"parameters":{"points":[[0,0],[21,0],[0,1]],"vertex":3,"frame":"orthogonal_equal_unit","operation":"density13_triangle_opposite_side_squares","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-opposite-side-squares/[3,\"441\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0047","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], [22, 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], 484)","explanation_zh":"對邊端點與平方距離=([2, 3], 484)。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":3,"skills":["標記頂點的對邊端點與平方"]},"variant":{"family_id":"qf-density13-triangle-opposite-side-squares","seed":202610137047,"parameters":{"points":[[0,1],[0,0],[22,0]],"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,\"484\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0048","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, 0], [0, 1], [23, 0]]。已要求三點相異且非共線，vertex=2。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":"([1, 3], 529)","explanation_zh":"對邊端點與平方距離=([1, 3], 529)。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":3,"skills":["標記頂點的對邊端點與平方"]},"variant":{"family_id":"qf-density13-triangle-opposite-side-squares","seed":202610137048,"parameters":{"points":[[0,0],[0,1],[23,0]],"vertex":2,"frame":"orthogonal_equal_unit","operation":"density13_triangle_opposite_side_squares","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-opposite-side-squares/[2,\"529\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0049","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, 0], [24, 0], [0, 1]]。已要求三點相異且非共線，vertex=3。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":"([1, 2], 576)","explanation_zh":"對邊端點與平方距離=([1, 2], 576)。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":3,"skills":["標記頂點的對邊端點與平方"]},"variant":{"family_id":"qf-density13-triangle-opposite-side-squares","seed":202610137049,"parameters":{"points":[[0,0],[24,0],[0,1]],"vertex":3,"frame":"orthogonal_equal_unit","operation":"density13_triangle_opposite_side_squares","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-opposite-side-squares/[3,\"576\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0050","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], [25, 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], 625)","explanation_zh":"對邊端點與平方距離=([2, 3], 625)。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":3,"skills":["標記頂點的對邊端點與平方"]},"variant":{"family_id":"qf-density13-triangle-opposite-side-squares","seed":202610137050,"parameters":{"points":[[0,1],[0,0],[25,0]],"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,\"625\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0051","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"三種頂點標記的行列式","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, 0], [1, 0], [0, -12]]。三點相異且非共線。依序求 (crossABC, crossBCA, crossACB) 有號整數，不把循環與交換標記混同。","answer":{"kind":"expression","value":"(-12, -12, 12)","explanation_zh":"三種標記座標行列式=(-12, -12, 12)。各標記次序分別套鞋帶行列式，循環與對換保留不同方向；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":["三種頂點標記的行列式"]},"variant":{"family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137051,"parameters":{"points":[[0,0],[1,0],[0,-12]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_cyclic_label_orientation","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"-12\""}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0052","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"三種頂點標記的行列式","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, 0], [1, 0], [0, -11]]。三點相異且非共線。依序求 (crossABC, crossBCA, crossACB) 有號整數，不把循環與交換標記混同。","answer":{"kind":"expression","value":"(-11, -11, 11)","explanation_zh":"三種標記座標行列式=(-11, -11, 11)。各標記次序分別套鞋帶行列式，循環與對換保留不同方向；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":["三種頂點標記的行列式"]},"variant":{"family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137052,"parameters":{"points":[[0,0],[1,0],[0,-11]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_cyclic_label_orientation","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"-11\""}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0053","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"三種頂點標記的行列式","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, 0], [1, 0], [0, -10]]。三點相異且非共線。依序求 (crossABC, crossBCA, crossACB) 有號整數，不把循環與交換標記混同。","answer":{"kind":"expression","value":"(-10, -10, 10)","explanation_zh":"三種標記座標行列式=(-10, -10, 10)。各標記次序分別套鞋帶行列式，循環與對換保留不同方向；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":["三種頂點標記的行列式"]},"variant":{"family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137053,"parameters":{"points":[[0,0],[1,0],[0,-10]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_cyclic_label_orientation","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"-10\""}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0054","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"三種頂點標記的行列式","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, 0], [1, 0], [0, -9]]。三點相異且非共線。依序求 (crossABC, crossBCA, crossACB) 有號整數，不把循環與交換標記混同。","answer":{"kind":"expression","value":"(-9, -9, 9)","explanation_zh":"三種標記座標行列式=(-9, -9, 9)。各標記次序分別套鞋帶行列式，循環與對換保留不同方向；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":["三種頂點標記的行列式"]},"variant":{"family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137054,"parameters":{"points":[[0,0],[1,0],[0,-9]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_cyclic_label_orientation","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"-9\""}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0055","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"三種頂點標記的行列式","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, 0], [1, 0], [0, -8]]。三點相異且非共線。依序求 (crossABC, crossBCA, crossACB) 有號整數，不把循環與交換標記混同。","answer":{"kind":"expression","value":"(-8, -8, 8)","explanation_zh":"三種標記座標行列式=(-8, -8, 8)。各標記次序分別套鞋帶行列式，循環與對換保留不同方向；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":["三種頂點標記的行列式"]},"variant":{"family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137055,"parameters":{"points":[[0,0],[1,0],[0,-8]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_cyclic_label_orientation","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"-8\""}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0056","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"三種頂點標記的行列式","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, 0], [1, 0], [0, -7]]。三點相異且非共線。依序求 (crossABC, crossBCA, crossACB) 有號整數，不把循環與交換標記混同。","answer":{"kind":"expression","value":"(-7, -7, 7)","explanation_zh":"三種標記座標行列式=(-7, -7, 7)。各標記次序分別套鞋帶行列式，循環與對換保留不同方向；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":["三種頂點標記的行列式"]},"variant":{"family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137056,"parameters":{"points":[[0,0],[1,0],[0,-7]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_cyclic_label_orientation","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"-7\""}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0057","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"三種頂點標記的行列式","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, 0], [1, 0], [0, -6]]。三點相異且非共線。依序求 (crossABC, crossBCA, crossACB) 有號整數，不把循環與交換標記混同。","answer":{"kind":"expression","value":"(-6, -6, 6)","explanation_zh":"三種標記座標行列式=(-6, -6, 6)。各標記次序分別套鞋帶行列式，循環與對換保留不同方向；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":["三種頂點標記的行列式"]},"variant":{"family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137057,"parameters":{"points":[[0,0],[1,0],[0,-6]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_cyclic_label_orientation","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"-6\""}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0058","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"三種頂點標記的行列式","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, 0], [1, 0], [0, -5]]。三點相異且非共線。依序求 (crossABC, crossBCA, crossACB) 有號整數，不把循環與交換標記混同。","answer":{"kind":"expression","value":"(-5, -5, 5)","explanation_zh":"三種標記座標行列式=(-5, -5, 5)。各標記次序分別套鞋帶行列式，循環與對換保留不同方向；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":["三種頂點標記的行列式"]},"variant":{"family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137058,"parameters":{"points":[[0,0],[1,0],[0,-5]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_cyclic_label_orientation","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"-5\""}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0059","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"三種頂點標記的行列式","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, 0], [1, 0], [0, -4]]。三點相異且非共線。依序求 (crossABC, crossBCA, crossACB) 有號整數，不把循環與交換標記混同。","answer":{"kind":"expression","value":"(-4, -4, 4)","explanation_zh":"三種標記座標行列式=(-4, -4, 4)。各標記次序分別套鞋帶行列式，循環與對換保留不同方向；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":2,"skills":["三種頂點標記的行列式"]},"variant":{"family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137059,"parameters":{"points":[[0,0],[1,0],[0,-4]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_cyclic_label_orientation","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"-4\""}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0060","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"三種頂點標記的行列式","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, 0], [1, 0], [0, -3]]。三點相異且非共線。依序求 (crossABC, crossBCA, crossACB) 有號整數，不把循環與交換標記混同。","answer":{"kind":"expression","value":"(-3, -3, 3)","explanation_zh":"三種標記座標行列式=(-3, -3, 3)。各標記次序分別套鞋帶行列式，循環與對換保留不同方向；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":2,"skills":["三種頂點標記的行列式"]},"variant":{"family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137060,"parameters":{"points":[[0,0],[1,0],[0,-3]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_cyclic_label_orientation","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"-3\""}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0061","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"三種頂點標記的行列式","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, 0], [1, 0], [0, -2]]。三點相異且非共線。依序求 (crossABC, crossBCA, crossACB) 有號整數，不把循環與交換標記混同。","answer":{"kind":"expression","value":"(-2, -2, 2)","explanation_zh":"三種標記座標行列式=(-2, -2, 2)。各標記次序分別套鞋帶行列式，循環與對換保留不同方向；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":2,"skills":["三種頂點標記的行列式"]},"variant":{"family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137061,"parameters":{"points":[[0,0],[1,0],[0,-2]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_cyclic_label_orientation","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"-2\""}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0062","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"三種頂點標記的行列式","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, 0], [1, 0], [0, -1]]。三點相異且非共線。依序求 (crossABC, crossBCA, crossACB) 有號整數，不把循環與交換標記混同。","answer":{"kind":"expression","value":"(-1, -1, 1)","explanation_zh":"三種標記座標行列式=(-1, -1, 1)。各標記次序分別套鞋帶行列式，循環與對換保留不同方向；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":2,"skills":["三種頂點標記的行列式"]},"variant":{"family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137062,"parameters":{"points":[[0,0],[1,0],[0,-1]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_cyclic_label_orientation","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"-1\""}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0063","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"三種頂點標記的行列式","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, 0], [1, 0], [0, 1]]。三點相異且非共線。依序求 (crossABC, crossBCA, crossACB) 有號整數，不把循環與交換標記混同。","answer":{"kind":"expression","value":"(1, 1, -1)","explanation_zh":"三種標記座標行列式=(1, 1, -1)。各標記次序分別套鞋帶行列式，循環與對換保留不同方向；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":2,"skills":["三種頂點標記的行列式"]},"variant":{"family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137063,"parameters":{"points":[[0,0],[1,0],[0,1]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_cyclic_label_orientation","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"1\""}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0064","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"三種頂點標記的行列式","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, 0], [1, 0], [0, 2]]。三點相異且非共線。依序求 (crossABC, crossBCA, crossACB) 有號整數，不把循環與交換標記混同。","answer":{"kind":"expression","value":"(2, 2, -2)","explanation_zh":"三種標記座標行列式=(2, 2, -2)。各標記次序分別套鞋帶行列式，循環與對換保留不同方向；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":2,"skills":["三種頂點標記的行列式"]},"variant":{"family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137064,"parameters":{"points":[[0,0],[1,0],[0,2]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_cyclic_label_orientation","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"2\""}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0065","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"三種頂點標記的行列式","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, 0], [1, 0], [0, 3]]。三點相異且非共線。依序求 (crossABC, crossBCA, crossACB) 有號整數，不把循環與交換標記混同。","answer":{"kind":"expression","value":"(3, 3, -3)","explanation_zh":"三種標記座標行列式=(3, 3, -3)。各標記次序分別套鞋帶行列式，循環與對換保留不同方向；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":2,"skills":["三種頂點標記的行列式"]},"variant":{"family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137065,"parameters":{"points":[[0,0],[1,0],[0,3]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_cyclic_label_orientation","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"3\""}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0066","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"三種頂點標記的行列式","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, 0], [1, 0], [0, 4]]。三點相異且非共線。依序求 (crossABC, crossBCA, crossACB) 有號整數，不把循環與交換標記混同。","answer":{"kind":"expression","value":"(4, 4, -4)","explanation_zh":"三種標記座標行列式=(4, 4, -4)。各標記次序分別套鞋帶行列式，循環與對換保留不同方向；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":2,"skills":["三種頂點標記的行列式"]},"variant":{"family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137066,"parameters":{"points":[[0,0],[1,0],[0,4]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_cyclic_label_orientation","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"4\""}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0067","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"三種頂點標記的行列式","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, 0], [1, 0], [0, 5]]。三點相異且非共線。依序求 (crossABC, crossBCA, crossACB) 有號整數，不把循環與交換標記混同。","answer":{"kind":"expression","value":"(5, 5, -5)","explanation_zh":"三種標記座標行列式=(5, 5, -5)。各標記次序分別套鞋帶行列式，循環與對換保留不同方向；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":2,"skills":["三種頂點標記的行列式"]},"variant":{"family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137067,"parameters":{"points":[[0,0],[1,0],[0,5]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_cyclic_label_orientation","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"5\""}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0068","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"三種頂點標記的行列式","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, 0], [1, 0], [0, 6]]。三點相異且非共線。依序求 (crossABC, crossBCA, crossACB) 有號整數，不把循環與交換標記混同。","answer":{"kind":"expression","value":"(6, 6, -6)","explanation_zh":"三種標記座標行列式=(6, 6, -6)。各標記次序分別套鞋帶行列式，循環與對換保留不同方向；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":3,"skills":["三種頂點標記的行列式"]},"variant":{"family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137068,"parameters":{"points":[[0,0],[1,0],[0,6]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_cyclic_label_orientation","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"6\""}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0069","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"三種頂點標記的行列式","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, 0], [1, 0], [0, 7]]。三點相異且非共線。依序求 (crossABC, crossBCA, crossACB) 有號整數，不把循環與交換標記混同。","answer":{"kind":"expression","value":"(7, 7, -7)","explanation_zh":"三種標記座標行列式=(7, 7, -7)。各標記次序分別套鞋帶行列式，循環與對換保留不同方向；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":3,"skills":["三種頂點標記的行列式"]},"variant":{"family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137069,"parameters":{"points":[[0,0],[1,0],[0,7]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_cyclic_label_orientation","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"7\""}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0070","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"三種頂點標記的行列式","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, 0], [1, 0], [0, 8]]。三點相異且非共線。依序求 (crossABC, crossBCA, crossACB) 有號整數，不把循環與交換標記混同。","answer":{"kind":"expression","value":"(8, 8, -8)","explanation_zh":"三種標記座標行列式=(8, 8, -8)。各標記次序分別套鞋帶行列式，循環與對換保留不同方向；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":3,"skills":["三種頂點標記的行列式"]},"variant":{"family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137070,"parameters":{"points":[[0,0],[1,0],[0,8]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_cyclic_label_orientation","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"8\""}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0071","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"三種頂點標記的行列式","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, 0], [1, 0], [0, 9]]。三點相異且非共線。依序求 (crossABC, crossBCA, crossACB) 有號整數，不把循環與交換標記混同。","answer":{"kind":"expression","value":"(9, 9, -9)","explanation_zh":"三種標記座標行列式=(9, 9, -9)。各標記次序分別套鞋帶行列式，循環與對換保留不同方向；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":3,"skills":["三種頂點標記的行列式"]},"variant":{"family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137071,"parameters":{"points":[[0,0],[1,0],[0,9]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_cyclic_label_orientation","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"9\""}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0072","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"三種頂點標記的行列式","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, 0], [1, 0], [0, 10]]。三點相異且非共線。依序求 (crossABC, crossBCA, crossACB) 有號整數，不把循環與交換標記混同。","answer":{"kind":"expression","value":"(10, 10, -10)","explanation_zh":"三種標記座標行列式=(10, 10, -10)。各標記次序分別套鞋帶行列式，循環與對換保留不同方向；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":3,"skills":["三種頂點標記的行列式"]},"variant":{"family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137072,"parameters":{"points":[[0,0],[1,0],[0,10]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_cyclic_label_orientation","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"10\""}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0073","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"三種頂點標記的行列式","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, 0], [1, 0], [0, 11]]。三點相異且非共線。依序求 (crossABC, crossBCA, crossACB) 有號整數，不把循環與交換標記混同。","answer":{"kind":"expression","value":"(11, 11, -11)","explanation_zh":"三種標記座標行列式=(11, 11, -11)。各標記次序分別套鞋帶行列式，循環與對換保留不同方向；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":3,"skills":["三種頂點標記的行列式"]},"variant":{"family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137073,"parameters":{"points":[[0,0],[1,0],[0,11]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_cyclic_label_orientation","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"11\""}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0074","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"三種頂點標記的行列式","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, 0], [1, 0], [0, 12]]。三點相異且非共線。依序求 (crossABC, crossBCA, crossACB) 有號整數，不把循環與交換標記混同。","answer":{"kind":"expression","value":"(12, 12, -12)","explanation_zh":"三種標記座標行列式=(12, 12, -12)。各標記次序分別套鞋帶行列式，循環與對換保留不同方向；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":3,"skills":["三種頂點標記的行列式"]},"variant":{"family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137074,"parameters":{"points":[[0,0],[1,0],[0,12]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_cyclic_label_orientation","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"12\""}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0075","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"三種頂點標記的行列式","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, 0], [1, 0], [0, 13]]。三點相異且非共線。依序求 (crossABC, crossBCA, crossACB) 有號整數，不把循環與交換標記混同。","answer":{"kind":"expression","value":"(13, 13, -13)","explanation_zh":"三種標記座標行列式=(13, 13, -13)。各標記次序分別套鞋帶行列式，循環與對換保留不同方向；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":3,"skills":["三種頂點標記的行列式"]},"variant":{"family_id":"qf-density13-triangle-cyclic-label-orientation","seed":202610137075,"parameters":{"points":[[0,0],[1,0],[0,13]],"frame":"orthogonal_equal_unit","operation":"density13_triangle_cyclic_label_orientation","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-cyclic-label-orientation/\"13\""}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0076","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"對邊中點與三段平方","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], [1, 0]]。三點相異且非共線，vertex=1，M為對邊中點。依序求 (Mx, My, 第一對邊端點至M平方, 第二端點至M平方, vertex至M平方)，五項均為最簡有理數。","answer":{"kind":"expression","value":"(1/2, 0/1, 1/4, 1/4, 5/4)","explanation_zh":"對邊中點與三段平方=(1/2, 0/1, 1/4, 1/4, 5/4)。座標和除二取得中點，用雙倍座標差平方除四求各段平方；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":["對邊中點與三段平方"]},"variant":{"family_id":"qf-density13-triangle-median-equality","seed":202610137076,"parameters":{"points":[[0,1],[0,0],[1,0]],"vertex":1,"frame":"orthogonal_equal_unit","operation":"density13_triangle_median_equality","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-median-equality/[1,[\"1/2\",\"0/1\"],\"1/4\",\"5/4\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0077","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"對邊中點與三段平方","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=[[-1, 0], [0, 2], [2, 0]]。三點相異且非共線，vertex=2，M為對邊中點。依序求 (Mx, My, 第一對邊端點至M平方, 第二端點至M平方, vertex至M平方)，五項均為最簡有理數。","answer":{"kind":"expression","value":"(1/2, 0/1, 9/4, 9/4, 17/4)","explanation_zh":"對邊中點與三段平方=(1/2, 0/1, 9/4, 9/4, 17/4)。座標和除二取得中點，用雙倍座標差平方除四求各段平方；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":["對邊中點與三段平方"]},"variant":{"family_id":"qf-density13-triangle-median-equality","seed":202610137077,"parameters":{"points":[[-1,0],[0,2],[2,0]],"vertex":2,"frame":"orthogonal_equal_unit","operation":"density13_triangle_median_equality","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-median-equality/[2,[\"1/2\",\"0/1\"],\"9/4\",\"17/4\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0078","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"對邊中點與三段平方","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=[[-2, 0], [3, 0], [0, 3]]。三點相異且非共線，vertex=3，M為對邊中點。依序求 (Mx, My, 第一對邊端點至M平方, 第二端點至M平方, vertex至M平方)，五項均為最簡有理數。","answer":{"kind":"expression","value":"(1/2, 0/1, 25/4, 25/4, 37/4)","explanation_zh":"對邊中點與三段平方=(1/2, 0/1, 25/4, 25/4, 37/4)。座標和除二取得中點，用雙倍座標差平方除四求各段平方；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":["對邊中點與三段平方"]},"variant":{"family_id":"qf-density13-triangle-median-equality","seed":202610137078,"parameters":{"points":[[-2,0],[3,0],[0,3]],"vertex":3,"frame":"orthogonal_equal_unit","operation":"density13_triangle_median_equality","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-median-equality/[3,[\"1/2\",\"0/1\"],\"25/4\",\"37/4\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0079","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"對邊中點與三段平方","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, 4], [-3, 0], [4, 0]]。三點相異且非共線，vertex=1，M為對邊中點。依序求 (Mx, My, 第一對邊端點至M平方, 第二端點至M平方, vertex至M平方)，五項均為最簡有理數。","answer":{"kind":"expression","value":"(1/2, 0/1, 49/4, 49/4, 65/4)","explanation_zh":"對邊中點與三段平方=(1/2, 0/1, 49/4, 49/4, 65/4)。座標和除二取得中點，用雙倍座標差平方除四求各段平方；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":["對邊中點與三段平方"]},"variant":{"family_id":"qf-density13-triangle-median-equality","seed":202610137079,"parameters":{"points":[[0,4],[-3,0],[4,0]],"vertex":1,"frame":"orthogonal_equal_unit","operation":"density13_triangle_median_equality","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-median-equality/[1,[\"1/2\",\"0/1\"],\"49/4\",\"65/4\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0080","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"對邊中點與三段平方","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=[[-4, 0], [0, 5], [5, 0]]。三點相異且非共線，vertex=2，M為對邊中點。依序求 (Mx, My, 第一對邊端點至M平方, 第二端點至M平方, vertex至M平方)，五項均為最簡有理數。","answer":{"kind":"expression","value":"(1/2, 0/1, 81/4, 81/4, 101/4)","explanation_zh":"對邊中點與三段平方=(1/2, 0/1, 81/4, 81/4, 101/4)。座標和除二取得中點，用雙倍座標差平方除四求各段平方；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":["對邊中點與三段平方"]},"variant":{"family_id":"qf-density13-triangle-median-equality","seed":202610137080,"parameters":{"points":[[-4,0],[0,5],[5,0]],"vertex":2,"frame":"orthogonal_equal_unit","operation":"density13_triangle_median_equality","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-median-equality/[2,[\"1/2\",\"0/1\"],\"81/4\",\"101/4\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0081","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"對邊中點與三段平方","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=[[-5, 0], [6, 0], [0, 6]]。三點相異且非共線，vertex=3，M為對邊中點。依序求 (Mx, My, 第一對邊端點至M平方, 第二端點至M平方, vertex至M平方)，五項均為最簡有理數。","answer":{"kind":"expression","value":"(1/2, 0/1, 121/4, 121/4, 145/4)","explanation_zh":"對邊中點與三段平方=(1/2, 0/1, 121/4, 121/4, 145/4)。座標和除二取得中點，用雙倍座標差平方除四求各段平方；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":["對邊中點與三段平方"]},"variant":{"family_id":"qf-density13-triangle-median-equality","seed":202610137081,"parameters":{"points":[[-5,0],[6,0],[0,6]],"vertex":3,"frame":"orthogonal_equal_unit","operation":"density13_triangle_median_equality","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-median-equality/[3,[\"1/2\",\"0/1\"],\"121/4\",\"145/4\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0082","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"對邊中點與三段平方","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, 7], [-6, 0], [7, 0]]。三點相異且非共線，vertex=1，M為對邊中點。依序求 (Mx, My, 第一對邊端點至M平方, 第二端點至M平方, vertex至M平方)，五項均為最簡有理數。","answer":{"kind":"expression","value":"(1/2, 0/1, 169/4, 169/4, 197/4)","explanation_zh":"對邊中點與三段平方=(1/2, 0/1, 169/4, 169/4, 197/4)。座標和除二取得中點，用雙倍座標差平方除四求各段平方；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":["對邊中點與三段平方"]},"variant":{"family_id":"qf-density13-triangle-median-equality","seed":202610137082,"parameters":{"points":[[0,7],[-6,0],[7,0]],"vertex":1,"frame":"orthogonal_equal_unit","operation":"density13_triangle_median_equality","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-median-equality/[1,[\"1/2\",\"0/1\"],\"169/4\",\"197/4\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0083","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"對邊中點與三段平方","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=[[-7, 0], [0, 8], [8, 0]]。三點相異且非共線，vertex=2，M為對邊中點。依序求 (Mx, My, 第一對邊端點至M平方, 第二端點至M平方, vertex至M平方)，五項均為最簡有理數。","answer":{"kind":"expression","value":"(1/2, 0/1, 225/4, 225/4, 257/4)","explanation_zh":"對邊中點與三段平方=(1/2, 0/1, 225/4, 225/4, 257/4)。座標和除二取得中點，用雙倍座標差平方除四求各段平方；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":["對邊中點與三段平方"]},"variant":{"family_id":"qf-density13-triangle-median-equality","seed":202610137083,"parameters":{"points":[[-7,0],[0,8],[8,0]],"vertex":2,"frame":"orthogonal_equal_unit","operation":"density13_triangle_median_equality","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-median-equality/[2,[\"1/2\",\"0/1\"],\"225/4\",\"257/4\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0084","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"對邊中點與三段平方","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=[[-8, 0], [9, 0], [0, 9]]。三點相異且非共線，vertex=3，M為對邊中點。依序求 (Mx, My, 第一對邊端點至M平方, 第二端點至M平方, vertex至M平方)，五項均為最簡有理數。","answer":{"kind":"expression","value":"(1/2, 0/1, 289/4, 289/4, 325/4)","explanation_zh":"對邊中點與三段平方=(1/2, 0/1, 289/4, 289/4, 325/4)。座標和除二取得中點，用雙倍座標差平方除四求各段平方；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":2,"skills":["對邊中點與三段平方"]},"variant":{"family_id":"qf-density13-triangle-median-equality","seed":202610137084,"parameters":{"points":[[-8,0],[9,0],[0,9]],"vertex":3,"frame":"orthogonal_equal_unit","operation":"density13_triangle_median_equality","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-median-equality/[3,[\"1/2\",\"0/1\"],\"289/4\",\"325/4\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0085","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"對邊中點與三段平方","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, 10], [-9, 0], [10, 0]]。三點相異且非共線，vertex=1，M為對邊中點。依序求 (Mx, My, 第一對邊端點至M平方, 第二端點至M平方, vertex至M平方)，五項均為最簡有理數。","answer":{"kind":"expression","value":"(1/2, 0/1, 361/4, 361/4, 401/4)","explanation_zh":"對邊中點與三段平方=(1/2, 0/1, 361/4, 361/4, 401/4)。座標和除二取得中點，用雙倍座標差平方除四求各段平方；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":2,"skills":["對邊中點與三段平方"]},"variant":{"family_id":"qf-density13-triangle-median-equality","seed":202610137085,"parameters":{"points":[[0,10],[-9,0],[10,0]],"vertex":1,"frame":"orthogonal_equal_unit","operation":"density13_triangle_median_equality","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-median-equality/[1,[\"1/2\",\"0/1\"],\"361/4\",\"401/4\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0086","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"對邊中點與三段平方","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=[[-10, 0], [0, 11], [11, 0]]。三點相異且非共線，vertex=2，M為對邊中點。依序求 (Mx, My, 第一對邊端點至M平方, 第二端點至M平方, vertex至M平方)，五項均為最簡有理數。","answer":{"kind":"expression","value":"(1/2, 0/1, 441/4, 441/4, 485/4)","explanation_zh":"對邊中點與三段平方=(1/2, 0/1, 441/4, 441/4, 485/4)。座標和除二取得中點，用雙倍座標差平方除四求各段平方；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":2,"skills":["對邊中點與三段平方"]},"variant":{"family_id":"qf-density13-triangle-median-equality","seed":202610137086,"parameters":{"points":[[-10,0],[0,11],[11,0]],"vertex":2,"frame":"orthogonal_equal_unit","operation":"density13_triangle_median_equality","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-median-equality/[2,[\"1/2\",\"0/1\"],\"441/4\",\"485/4\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0087","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"對邊中點與三段平方","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=[[-11, 0], [12, 0], [0, 12]]。三點相異且非共線，vertex=3，M為對邊中點。依序求 (Mx, My, 第一對邊端點至M平方, 第二端點至M平方, vertex至M平方)，五項均為最簡有理數。","answer":{"kind":"expression","value":"(1/2, 0/1, 529/4, 529/4, 577/4)","explanation_zh":"對邊中點與三段平方=(1/2, 0/1, 529/4, 529/4, 577/4)。座標和除二取得中點，用雙倍座標差平方除四求各段平方；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":2,"skills":["對邊中點與三段平方"]},"variant":{"family_id":"qf-density13-triangle-median-equality","seed":202610137087,"parameters":{"points":[[-11,0],[12,0],[0,12]],"vertex":3,"frame":"orthogonal_equal_unit","operation":"density13_triangle_median_equality","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-median-equality/[3,[\"1/2\",\"0/1\"],\"529/4\",\"577/4\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0088","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"對邊中點與三段平方","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, 13], [-12, 0], [13, 0]]。三點相異且非共線，vertex=1，M為對邊中點。依序求 (Mx, My, 第一對邊端點至M平方, 第二端點至M平方, vertex至M平方)，五項均為最簡有理數。","answer":{"kind":"expression","value":"(1/2, 0/1, 625/4, 625/4, 677/4)","explanation_zh":"對邊中點與三段平方=(1/2, 0/1, 625/4, 625/4, 677/4)。座標和除二取得中點，用雙倍座標差平方除四求各段平方；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":2,"skills":["對邊中點與三段平方"]},"variant":{"family_id":"qf-density13-triangle-median-equality","seed":202610137088,"parameters":{"points":[[0,13],[-12,0],[13,0]],"vertex":1,"frame":"orthogonal_equal_unit","operation":"density13_triangle_median_equality","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-median-equality/[1,[\"1/2\",\"0/1\"],\"625/4\",\"677/4\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0089","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"對邊中點與三段平方","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=[[-13, 0], [0, 14], [14, 0]]。三點相異且非共線，vertex=2，M為對邊中點。依序求 (Mx, My, 第一對邊端點至M平方, 第二端點至M平方, vertex至M平方)，五項均為最簡有理數。","answer":{"kind":"expression","value":"(1/2, 0/1, 729/4, 729/4, 785/4)","explanation_zh":"對邊中點與三段平方=(1/2, 0/1, 729/4, 729/4, 785/4)。座標和除二取得中點，用雙倍座標差平方除四求各段平方；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":2,"skills":["對邊中點與三段平方"]},"variant":{"family_id":"qf-density13-triangle-median-equality","seed":202610137089,"parameters":{"points":[[-13,0],[0,14],[14,0]],"vertex":2,"frame":"orthogonal_equal_unit","operation":"density13_triangle_median_equality","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-median-equality/[2,[\"1/2\",\"0/1\"],\"729/4\",\"785/4\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0090","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"對邊中點與三段平方","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=[[-14, 0], [15, 0], [0, 15]]。三點相異且非共線，vertex=3，M為對邊中點。依序求 (Mx, My, 第一對邊端點至M平方, 第二端點至M平方, vertex至M平方)，五項均為最簡有理數。","answer":{"kind":"expression","value":"(1/2, 0/1, 841/4, 841/4, 901/4)","explanation_zh":"對邊中點與三段平方=(1/2, 0/1, 841/4, 841/4, 901/4)。座標和除二取得中點，用雙倍座標差平方除四求各段平方；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":2,"skills":["對邊中點與三段平方"]},"variant":{"family_id":"qf-density13-triangle-median-equality","seed":202610137090,"parameters":{"points":[[-14,0],[15,0],[0,15]],"vertex":3,"frame":"orthogonal_equal_unit","operation":"density13_triangle_median_equality","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-median-equality/[3,[\"1/2\",\"0/1\"],\"841/4\",\"901/4\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0091","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"對邊中點與三段平方","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, 16], [-15, 0], [16, 0]]。三點相異且非共線，vertex=1，M為對邊中點。依序求 (Mx, My, 第一對邊端點至M平方, 第二端點至M平方, vertex至M平方)，五項均為最簡有理數。","answer":{"kind":"expression","value":"(1/2, 0/1, 961/4, 961/4, 1025/4)","explanation_zh":"對邊中點與三段平方=(1/2, 0/1, 961/4, 961/4, 1025/4)。座標和除二取得中點，用雙倍座標差平方除四求各段平方；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":2,"skills":["對邊中點與三段平方"]},"variant":{"family_id":"qf-density13-triangle-median-equality","seed":202610137091,"parameters":{"points":[[0,16],[-15,0],[16,0]],"vertex":1,"frame":"orthogonal_equal_unit","operation":"density13_triangle_median_equality","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-median-equality/[1,[\"1/2\",\"0/1\"],\"961/4\",\"1025/4\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0092","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"對邊中點與三段平方","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=[[-16, 0], [0, 17], [17, 0]]。三點相異且非共線，vertex=2，M為對邊中點。依序求 (Mx, My, 第一對邊端點至M平方, 第二端點至M平方, vertex至M平方)，五項均為最簡有理數。","answer":{"kind":"expression","value":"(1/2, 0/1, 1089/4, 1089/4, 1157/4)","explanation_zh":"對邊中點與三段平方=(1/2, 0/1, 1089/4, 1089/4, 1157/4)。座標和除二取得中點，用雙倍座標差平方除四求各段平方；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":2,"skills":["對邊中點與三段平方"]},"variant":{"family_id":"qf-density13-triangle-median-equality","seed":202610137092,"parameters":{"points":[[-16,0],[0,17],[17,0]],"vertex":2,"frame":"orthogonal_equal_unit","operation":"density13_triangle_median_equality","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-median-equality/[2,[\"1/2\",\"0/1\"],\"1089/4\",\"1157/4\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0093","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"對邊中點與三段平方","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=[[-17, 0], [18, 0], [0, 18]]。三點相異且非共線，vertex=3，M為對邊中點。依序求 (Mx, My, 第一對邊端點至M平方, 第二端點至M平方, vertex至M平方)，五項均為最簡有理數。","answer":{"kind":"expression","value":"(1/2, 0/1, 1225/4, 1225/4, 1297/4)","explanation_zh":"對邊中點與三段平方=(1/2, 0/1, 1225/4, 1225/4, 1297/4)。座標和除二取得中點，用雙倍座標差平方除四求各段平方；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":3,"skills":["對邊中點與三段平方"]},"variant":{"family_id":"qf-density13-triangle-median-equality","seed":202610137093,"parameters":{"points":[[-17,0],[18,0],[0,18]],"vertex":3,"frame":"orthogonal_equal_unit","operation":"density13_triangle_median_equality","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-median-equality/[3,[\"1/2\",\"0/1\"],\"1225/4\",\"1297/4\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0094","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"對邊中點與三段平方","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, 19], [-18, 0], [19, 0]]。三點相異且非共線，vertex=1，M為對邊中點。依序求 (Mx, My, 第一對邊端點至M平方, 第二端點至M平方, vertex至M平方)，五項均為最簡有理數。","answer":{"kind":"expression","value":"(1/2, 0/1, 1369/4, 1369/4, 1445/4)","explanation_zh":"對邊中點與三段平方=(1/2, 0/1, 1369/4, 1369/4, 1445/4)。座標和除二取得中點，用雙倍座標差平方除四求各段平方；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":3,"skills":["對邊中點與三段平方"]},"variant":{"family_id":"qf-density13-triangle-median-equality","seed":202610137094,"parameters":{"points":[[0,19],[-18,0],[19,0]],"vertex":1,"frame":"orthogonal_equal_unit","operation":"density13_triangle_median_equality","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-median-equality/[1,[\"1/2\",\"0/1\"],\"1369/4\",\"1445/4\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0095","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"對邊中點與三段平方","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=[[-19, 0], [0, 20], [20, 0]]。三點相異且非共線，vertex=2，M為對邊中點。依序求 (Mx, My, 第一對邊端點至M平方, 第二端點至M平方, vertex至M平方)，五項均為最簡有理數。","answer":{"kind":"expression","value":"(1/2, 0/1, 1521/4, 1521/4, 1601/4)","explanation_zh":"對邊中點與三段平方=(1/2, 0/1, 1521/4, 1521/4, 1601/4)。座標和除二取得中點，用雙倍座標差平方除四求各段平方；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":3,"skills":["對邊中點與三段平方"]},"variant":{"family_id":"qf-density13-triangle-median-equality","seed":202610137095,"parameters":{"points":[[-19,0],[0,20],[20,0]],"vertex":2,"frame":"orthogonal_equal_unit","operation":"density13_triangle_median_equality","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-median-equality/[2,[\"1/2\",\"0/1\"],\"1521/4\",\"1601/4\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0096","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"對邊中點與三段平方","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=[[-20, 0], [21, 0], [0, 21]]。三點相異且非共線，vertex=3，M為對邊中點。依序求 (Mx, My, 第一對邊端點至M平方, 第二端點至M平方, vertex至M平方)，五項均為最簡有理數。","answer":{"kind":"expression","value":"(1/2, 0/1, 1681/4, 1681/4, 1765/4)","explanation_zh":"對邊中點與三段平方=(1/2, 0/1, 1681/4, 1681/4, 1765/4)。座標和除二取得中點，用雙倍座標差平方除四求各段平方；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":3,"skills":["對邊中點與三段平方"]},"variant":{"family_id":"qf-density13-triangle-median-equality","seed":202610137096,"parameters":{"points":[[-20,0],[21,0],[0,21]],"vertex":3,"frame":"orthogonal_equal_unit","operation":"density13_triangle_median_equality","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-median-equality/[3,[\"1/2\",\"0/1\"],\"1681/4\",\"1765/4\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0097","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"對邊中點與三段平方","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, 22], [-21, 0], [22, 0]]。三點相異且非共線，vertex=1，M為對邊中點。依序求 (Mx, My, 第一對邊端點至M平方, 第二端點至M平方, vertex至M平方)，五項均為最簡有理數。","answer":{"kind":"expression","value":"(1/2, 0/1, 1849/4, 1849/4, 1937/4)","explanation_zh":"對邊中點與三段平方=(1/2, 0/1, 1849/4, 1849/4, 1937/4)。座標和除二取得中點，用雙倍座標差平方除四求各段平方；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":3,"skills":["對邊中點與三段平方"]},"variant":{"family_id":"qf-density13-triangle-median-equality","seed":202610137097,"parameters":{"points":[[0,22],[-21,0],[22,0]],"vertex":1,"frame":"orthogonal_equal_unit","operation":"density13_triangle_median_equality","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-median-equality/[1,[\"1/2\",\"0/1\"],\"1849/4\",\"1937/4\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0098","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"對邊中點與三段平方","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=[[-22, 0], [0, 23], [23, 0]]。三點相異且非共線，vertex=2，M為對邊中點。依序求 (Mx, My, 第一對邊端點至M平方, 第二端點至M平方, vertex至M平方)，五項均為最簡有理數。","answer":{"kind":"expression","value":"(1/2, 0/1, 2025/4, 2025/4, 2117/4)","explanation_zh":"對邊中點與三段平方=(1/2, 0/1, 2025/4, 2025/4, 2117/4)。座標和除二取得中點，用雙倍座標差平方除四求各段平方；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":3,"skills":["對邊中點與三段平方"]},"variant":{"family_id":"qf-density13-triangle-median-equality","seed":202610137098,"parameters":{"points":[[-22,0],[0,23],[23,0]],"vertex":2,"frame":"orthogonal_equal_unit","operation":"density13_triangle_median_equality","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-median-equality/[2,[\"1/2\",\"0/1\"],\"2025/4\",\"2117/4\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0099","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"對邊中點與三段平方","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=[[-23, 0], [24, 0], [0, 24]]。三點相異且非共線，vertex=3，M為對邊中點。依序求 (Mx, My, 第一對邊端點至M平方, 第二端點至M平方, vertex至M平方)，五項均為最簡有理數。","answer":{"kind":"expression","value":"(1/2, 0/1, 2209/4, 2209/4, 2305/4)","explanation_zh":"對邊中點與三段平方=(1/2, 0/1, 2209/4, 2209/4, 2305/4)。座標和除二取得中點，用雙倍座標差平方除四求各段平方；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":3,"skills":["對邊中點與三段平方"]},"variant":{"family_id":"qf-density13-triangle-median-equality","seed":202610137099,"parameters":{"points":[[-23,0],[24,0],[0,24]],"vertex":3,"frame":"orthogonal_equal_unit","operation":"density13_triangle_median_equality","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-median-equality/[3,[\"1/2\",\"0/1\"],\"2209/4\",\"2305/4\"]"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r07-triangle-0100","batch_id":"batch-20261009-advance-20261013-r07-triangle-foundations","language":"zh-Hant","source_mko_ids":["mko-triangle"],"title_zh":"對邊中點與三段平方","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, 25], [-24, 0], [25, 0]]。三點相異且非共線，vertex=1，M為對邊中點。依序求 (Mx, My, 第一對邊端點至M平方, 第二端點至M平方, vertex至M平方)，五項均為最簡有理數。","answer":{"kind":"expression","value":"(1/2, 0/1, 2401/4, 2401/4, 2501/4)","explanation_zh":"對邊中點與三段平方=(1/2, 0/1, 2401/4, 2401/4, 2501/4)。座標和除二取得中點，用雙倍座標差平方除四求各段平方；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":3,"skills":["對邊中點與三段平方"]},"variant":{"family_id":"qf-density13-triangle-median-equality","seed":202610137100,"parameters":{"points":[[0,25],[-24,0],[25,0]],"vertex":1,"frame":"orthogonal_equal_unit","operation":"density13_triangle_median_equality","domain":"finite_labelled_euclidean_triangle","semantic_template":"qf-density13-triangle-median-equality/[1,[\"1/2\",\"0/1\"],\"2401/4\",\"2501/4\"]"}},"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"}}