{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261006-advance-20261007-r05-congruence-0089",
  "pilot_number": "R0005-03314",
  "number": 3314,
  "family_id": "qf-congruence-triangle-validity",
  "title": {
    "zh-Hant": "第 3314 題：有序三點的三角形資格",
    "en": "Question 3314: Triangle Eligibility of an Ordered Point Triple"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261006-advance-20261007-r05-congruence-0089",
    "batch_id": "batch-20261006-advance-20261007-r05-congruence-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-congruence"
    ],
    "title_zh": "有序三點的三角形資格",
    "stem_zh": "等單位正交座標系，兩軸採同一長度單位。有序頂點 A=[[0,0],[13,0],[14,0]]。是否為相異不共線的非退化三角形？重複或共線資料回答否。",
    "answer": {
      "kind": "short_text",
      "value": "否",
      "explanation_zh": "共線判別行列式=0。用有序 shoelace 的非零判別式檢查，與圖形目測分開；不把退化資料拿去做 SSS。"
    },
    "difficulty": {
      "band": "lower_secondary",
      "level": 2,
      "skills": [
        "有序三點的三角形資格"
      ]
    },
    "variant": {
      "family_id": "qf-congruence-triangle-validity",
      "seed": 202610075089,
      "parameters": {
        "A": [
          [
            0,
            0
          ],
          [
            13,
            0
          ],
          [
            14,
            0
          ]
        ],
        "operation": "triangle_validity",
        "frame": "euclidean_orthonormal_equal_units",
        "semantic_template": "qf-congruence-triangle-validity/{\"inputs\":[\"1\",\"169\",\"196\"],\"operation\":\"triangle_validity\"}"
      }
    },
    "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": "2026-10-06",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261006-advance-20261007-r05-congruence-0089",
    "batch_id": "batch-20261006-advance-20261007-r05-congruence-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "96cc19c9dc7daedfa54366fcb3286a448df6cd4d268de6bb42344aa0ad5a05b0",
      "family_id": "qf-congruence-triangle-validity",
      "seed": 202610075089,
      "semantic_template": "qf-congruence-triangle-validity/{\"inputs\":[\"1\",\"169\",\"196\"],\"operation\":\"triangle_validity\"}"
    },
    "fields": {
      "title": "Triangle Eligibility of an Ordered Point Triple",
      "stem": "Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered points are A1=(0, 0), A2=(13, 0), A3=(14, 0). Do they form a nondegenerate triangle of distinct noncollinear points? Reject repeated or collinear data.",
      "answer": {
        "kind": "short_text",
        "value": "No",
        "explanation": "Test whether the determinant of the two differences A2−A1 and A3−A1 is nonzero, giving No. Keep this exact coordinate test separate from a visual guess, and do not apply SSS to a degenerate triple."
      },
      "skills": [
        "Triangle Eligibility of an Ordered Point Triple"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "等單位正交座標系，兩軸採同一長度單位。有序頂點 A=[[0,0],[13,0],[14,0]]。是否為相異不共線的非退化三角形？重複或共線資料回答否。",
        "en": "Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered points are A1=(0, 0), A2=(13, 0), A3=(14, 0). Do they form a nondegenerate triangle of distinct noncollinear points? Reject repeated or collinear data."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "ordered triples in an orthonormal equal-unit frame",
      "after": "vertex indices are fixed;allow rotations,translations and reflections",
      "relation": "implies",
      "rule_id": "fixed_vertex_correspondence",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按同索引對應，不自行重排頂點；歐氏全等允許反射。先檢查非退化，再用SSS。",
        "en": "Correspond equal indices without silently permuting vertices;Euclidean congruence allows reflection.Check nondegeneracy before SSS."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "A vector1 component0 := sub(13, 0)",
      "after": "A vector1 component0 = 13",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "從第一頂點相減得到基向量分量。",
        "en": "Subtract the first vertex to obtain an edge-vector coordinate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 A vector1 component0，不以小數近似替代。",
          "en": "Recompute A vector1 component0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "13",
          "0"
        ],
        "output": "13",
        "label": "A vector1 component0"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "A vector1 component1 := sub(0, 0)",
      "after": "A vector1 component1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "從第一頂點相減得到基向量分量。",
        "en": "Subtract the first vertex to obtain an edge-vector coordinate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 A vector1 component1，不以小數近似替代。",
          "en": "Recompute A vector1 component1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "A vector1 component1"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "A vector2 component0 := sub(14, 0)",
      "after": "A vector2 component0 = 14",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "從第一頂點相減得到基向量分量。",
        "en": "Subtract the first vertex to obtain an edge-vector coordinate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 A vector2 component0，不以小數近似替代。",
          "en": "Recompute A vector2 component0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "14",
          "0"
        ],
        "output": "14",
        "label": "A vector2 component0"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "A vector2 component1 := sub(0, 0)",
      "after": "A vector2 component1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "從第一頂點相減得到基向量分量。",
        "en": "Subtract the first vertex to obtain an edge-vector coordinate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 A vector2 component1，不以小數近似替代。",
          "en": "Recompute A vector2 component1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "A vector2 component1"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "A determinant positive term := mul(13, 0)",
      "after": "A determinant positive term = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "行列式先算第一個交叉乘積。",
        "en": "Compute the first cross product term of the determinant."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 A determinant positive term，不以小數近似替代。",
          "en": "Recompute A determinant positive term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "13",
          "0"
        ],
        "output": "0",
        "label": "A determinant positive term"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "A determinant negative term := mul(0, 14)",
      "after": "A determinant negative term = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第二個交叉乘積保留減號位置。",
        "en": "Compute the other cross product term,retaining its subtraction position."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 A determinant negative term，不以小數近似替代。",
          "en": "Recompute A determinant negative term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "14"
        ],
        "output": "0",
        "label": "A determinant negative term"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "A oriented double area := sub(0, 0)",
      "after": "A oriented double area = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩項相減；非零當且僅當三點不共線，也排除重複頂點。",
        "en": "Subtract the terms;nonzero means noncollinear and also excludes repeated vertices."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 A oriented double area，不以小數近似替代。",
          "en": "Recompute A oriented double area with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "A oriented double area"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "determinant A=0",
      "after": "proper triangle=否",
      "relation": "implies",
      "rule_id": "triangle_nondegeneracy",
      "conditions": [],
      "explanation": {
        "zh-Hant": "此精確行列式取代目測；退化三點不能被送入非退化三角形的SSS前提。",
        "en": "This exact determinant replaces visual guessing;degenerate triples do not meet the triangle premise of SSS."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": false,
    "determinant": "0"
  },
  "conclusion": {
    "canonical_value": "否",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
  "source": {
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      "english_rows_path": "public/data/i18n/questions/batch-20261006-advance-20261007-r05-congruence-foundations.en.jsonl"
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        "original_path": "public/data/mko/congruence.json",
        "english_path": "public/data/i18n/editions/mko-congruence.en.json"
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    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "66b95c054a23815e6aa9999f9605667abcb96e0c5a97b71cd67b67041065878a",
    "original_candidate_sha256": "8dd0935546dca3747608a384ffd21383356a494793ad715fa484fd67ee227ecb",
    "original_candidate_line_sha256": "8dd0935546dca3747608a384ffd21383356a494793ad715fa484fd67ee227ecb"
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  "historical_source_state": {
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      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-004-p6",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
    "review_receipts": {
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    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 9,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
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    "human_review": "NOT_PERFORMED",
    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
  }
}