{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261009-advance-20261013-r07-triangle-0088",
  "pilot_number": "R0012-06563",
  "number": 6563,
  "family_id": "qf-density13-triangle-median-equality",
  "title": {
    "zh-Hant": "第 6563 題：對邊中點與三段平方",
    "en": "Question 6563: Opposite Midpoint and Three Squared Segments"
  },
  "original_question": {
    "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"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261009-advance-20261013-r07-triangle-0088",
    "batch_id": "batch-20261009-advance-20261013-r07-triangle-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "8db0b38c8ac3f2490dd54372354d2cf04cd48f5d61329e06ce589d40f2856400",
      "family_id": "qf-density13-triangle-median-equality",
      "seed": 202610137088,
      "semantic_template": "qf-density13-triangle-median-equality/[1,[\"1/2\",\"0/1\"],\"625/4\",\"677/4\"]"
    },
    "fields": {
      "title": "Opposite Midpoint and Three Squared Segments",
      "stem": "In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 13], [-12, 0], [13, 0]]. The points are distinct and noncollinear, vertex=1, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.",
      "answer": {
        "kind": "expression",
        "value": "(1/2, 0/1, 625/4, 625/4, 677/4)",
        "explanation": "Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 625/4, 625/4, 677/4)."
      },
      "skills": [
        "Opposite Midpoint and Three Squared Segments"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "歐氏等單位正交座標；A、B、C按頂點索引1、2、3排列，points=[[0, 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。",
        "en": "In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 13], [-12, 0], [13, 0]]. The points are distinct and noncollinear, vertex=1, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "[[0, 13], [-12, 0], [13, 0]]",
      "after": "original vertex labels1,2,3 in an equal-unit orthogonal Euclidean frame",
      "relation": "implies",
      "rule_id": "labelled_triangle_coordinate_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留A、B、C原順序，等單位且正交才使用此座標平方公式；不由示意圖讀數，候選頂點未必是合格三角形。",
        "en": "Retain the original A,B,C order;the squared-coordinate formula uses equal units and orthogonal axes.Do not read values from a sketch or assume candidate vertices already form a triangle."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "[[0, 13], [-12, 0], [13, 0]]",
      "after": "three distinct points=是",
      "relation": "implies",
      "rule_id": "distinct_triangle_vertex_check",
      "conditions": [],
      "explanation": {
        "zh-Hant": "相異資格與非共線資格分開；三點共線可以彼此相異，重複頂點則不合非退化三角形。",
        "en": "Check distinctness separately from noncollinearity;collinear points may be distinct,while repeated vertices do not form a proper triangle."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "ABC first vector component0 := sub(-12, 0)",
      "after": "ABC first vector component0 = -12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第一基向量是第二頂點減第一頂點。",
        "en": "The first edge vector is the second vertex minus the first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ABC first vector component0，不以小數近似替代。",
          "en": "Recompute ABC first vector component0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-12",
          "0"
        ],
        "output": "-12",
        "label": "ABC first vector component0"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "ABC first vector component1 := sub(0, 13)",
      "after": "ABC first vector component1 = -13",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第一基向量是第二頂點減第一頂點。",
        "en": "The first edge vector is the second vertex minus the first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ABC first vector component1，不以小數近似替代。",
          "en": "Recompute ABC first vector component1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "13"
        ],
        "output": "-13",
        "label": "ABC first vector component1"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "ABC second vector component0 := sub(13, 0)",
      "after": "ABC second vector component0 = 13",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第二基向量是第三頂點減同一第一頂點。",
        "en": "The second edge vector is the third vertex minus that same first vertex."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ABC second vector component0，不以小數近似替代。",
          "en": "Recompute ABC second vector component0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "13",
          "0"
        ],
        "output": "13",
        "label": "ABC second vector component0"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "ABC second vector component1 := sub(0, 13)",
      "after": "ABC second vector component1 = -13",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第二基向量是第三頂點減同一第一頂點。",
        "en": "The second edge vector is the third vertex minus that same first vertex."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ABC second vector component1，不以小數近似替代。",
          "en": "Recompute ABC second vector component1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "13"
        ],
        "output": "-13",
        "label": "ABC second vector component1"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "ABC cross positive term := mul(-12, -13)",
      "after": "ABC cross positive term = 156",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先算u_x v_y。",
        "en": "Compute u_x v_y first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ABC cross positive term，不以小數近似替代。",
          "en": "Recompute ABC cross positive term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-12",
          "-13"
        ],
        "output": "156",
        "label": "ABC cross positive term"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "ABC cross negative term := mul(-13, 13)",
      "after": "ABC cross negative term = -169",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "另算u_y v_x，減號留到下一步。",
        "en": "Compute u_y v_x separately,retaining subtraction for the next step."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ABC cross negative term，不以小數近似替代。",
          "en": "Recompute ABC cross negative term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-13",
          "13"
        ],
        "output": "-169",
        "label": "ABC cross negative term"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "ABC signed determinant := sub(156, -169)",
      "after": "ABC signed determinant = 325",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "相減給有號二倍面積，不能取絕對值抹去標記方向。",
        "en": "Subtract to obtain signed double area;do not erase label orientation with an absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ABC signed determinant，不以小數近似替代。",
          "en": "Recompute ABC signed determinant with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "156",
          "-169"
        ],
        "output": "325",
        "label": "ABC signed determinant"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "distinct=是;signed determinant=325",
      "after": "proper triangle=是",
      "relation": "implies",
      "rule_id": "triangle_nondegenerate_coordinate_qualification",
      "conditions": [],
      "explanation": {
        "zh-Hant": "非退化要求相異且行列式非零；只對這組明示座標給判定，不冒稱一般幾何構造證明。",
        "en": "A proper triangle needs distinct vertices and nonzero determinant;this judges the supplied coordinates,not a general geometric construction proof."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "marked vertex=1",
      "after": "opposite endpoints=[2, 3]",
      "relation": "implies",
      "rule_id": "original_label_opposite_side",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對邊端點是另外兩個原標記，不依座標或邊長重新排序。",
        "en": "The opposite-side endpoints are the other two original labels,not a reordered coordinate or side-length list."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "midpoint coordinate sum0 := sum(-12, 13)",
      "after": "midpoint coordinate sum0 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對邊兩端的同分量先相加。",
        "en": "Add matching coordinates of the opposite endpoints."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 midpoint coordinate sum0，不以小數近似替代。",
          "en": "Recompute midpoint coordinate sum0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-12",
          "13"
        ],
        "output": "1",
        "label": "midpoint coordinate sum0"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "midpoint coordinate0 := div(1, 2)",
      "after": "midpoint coordinate0 = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整座標和除2，保留最簡有理值。",
        "en": "Divide the full sum by2,retaining the reduced rational value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 midpoint coordinate0，不以小數近似替代。",
          "en": "Recompute midpoint coordinate0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "1",
          "2"
        ],
        "output": "1/2",
        "label": "midpoint coordinate0"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "midpoint coordinate sum1 := sum(0, 0)",
      "after": "midpoint coordinate sum1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對邊兩端的同分量先相加。",
        "en": "Add matching coordinates of the opposite endpoints."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 midpoint coordinate sum1，不以小數近似替代。",
          "en": "Recompute midpoint coordinate sum1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "midpoint coordinate sum1"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "midpoint coordinate1 := div(0, 2)",
      "after": "midpoint coordinate1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整座標和除2，保留最簡有理值。",
        "en": "Divide the full sum by2,retaining the reduced rational value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 midpoint coordinate1，不以小數近似替代。",
          "en": "Recompute midpoint coordinate1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "0",
          "2"
        ],
        "output": "0",
        "label": "midpoint coordinate1"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "first endpoint to midpoint coordinate difference0 := sub(-12, 1/2)",
      "after": "first endpoint to midpoint coordinate difference0 = -25/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先取同分量座標差，允許中點有理數。",
        "en": "Subtract matching coordinates,allowing rational midpoint coordinates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 first endpoint to midpoint coordinate difference0，不以小數近似替代。",
          "en": "Recompute first endpoint to midpoint coordinate difference0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-12",
          "1/2"
        ],
        "output": "-25/2",
        "label": "first endpoint to midpoint coordinate difference0"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "first endpoint to midpoint difference squared0 := square(-25/2)",
      "after": "first endpoint to midpoint difference squared0 = 625/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐分量平方，差的正負不影響平方長度。",
        "en": "Square each difference;its sign does not affect squared length."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 first endpoint to midpoint difference squared0，不以小數近似替代。",
          "en": "Recompute first endpoint to midpoint difference squared0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "square",
        "inputs": [
          "-25/2"
        ],
        "output": "625/4",
        "label": "first endpoint to midpoint difference squared0"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "first endpoint to midpoint coordinate difference1 := sub(0, 0)",
      "after": "first endpoint to midpoint coordinate difference1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先取同分量座標差，允許中點有理數。",
        "en": "Subtract matching coordinates,allowing rational midpoint coordinates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 first endpoint to midpoint coordinate difference1，不以小數近似替代。",
          "en": "Recompute first endpoint to midpoint coordinate difference1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "first endpoint to midpoint coordinate difference1"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "first endpoint to midpoint difference squared1 := square(0)",
      "after": "first endpoint to midpoint difference squared1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐分量平方，差的正負不影響平方長度。",
        "en": "Square each difference;its sign does not affect squared length."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 first endpoint to midpoint difference squared1，不以小數近似替代。",
          "en": "Recompute first endpoint to midpoint difference squared1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "square",
        "inputs": [
          "0"
        ],
        "output": "0",
        "label": "first endpoint to midpoint difference squared1"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "first endpoint to midpoint squared distance := sum(625/4, 0)",
      "after": "first endpoint to midpoint squared distance = 625/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依等單位正交L2關係相加兩平方。",
        "en": "Add the two squares in the equal-unit orthogonal L2 frame."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 first endpoint to midpoint squared distance，不以小數近似替代。",
          "en": "Recompute first endpoint to midpoint squared distance with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "625/4",
          "0"
        ],
        "output": "625/4",
        "label": "first endpoint to midpoint squared distance"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "second endpoint to midpoint coordinate difference0 := sub(13, 1/2)",
      "after": "second endpoint to midpoint coordinate difference0 = 25/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先取同分量座標差，允許中點有理數。",
        "en": "Subtract matching coordinates,allowing rational midpoint coordinates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 second endpoint to midpoint coordinate difference0，不以小數近似替代。",
          "en": "Recompute second endpoint to midpoint coordinate difference0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "13",
          "1/2"
        ],
        "output": "25/2",
        "label": "second endpoint to midpoint coordinate difference0"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "second endpoint to midpoint difference squared0 := square(25/2)",
      "after": "second endpoint to midpoint difference squared0 = 625/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐分量平方，差的正負不影響平方長度。",
        "en": "Square each difference;its sign does not affect squared length."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 second endpoint to midpoint difference squared0，不以小數近似替代。",
          "en": "Recompute second endpoint to midpoint difference squared0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "square",
        "inputs": [
          "25/2"
        ],
        "output": "625/4",
        "label": "second endpoint to midpoint difference squared0"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "second endpoint to midpoint coordinate difference1 := sub(0, 0)",
      "after": "second endpoint to midpoint coordinate difference1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先取同分量座標差，允許中點有理數。",
        "en": "Subtract matching coordinates,allowing rational midpoint coordinates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 second endpoint to midpoint coordinate difference1，不以小數近似替代。",
          "en": "Recompute second endpoint to midpoint coordinate difference1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "second endpoint to midpoint coordinate difference1"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "second endpoint to midpoint difference squared1 := square(0)",
      "after": "second endpoint to midpoint difference squared1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐分量平方，差的正負不影響平方長度。",
        "en": "Square each difference;its sign does not affect squared length."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 second endpoint to midpoint difference squared1，不以小數近似替代。",
          "en": "Recompute second endpoint to midpoint difference squared1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "square",
        "inputs": [
          "0"
        ],
        "output": "0",
        "label": "second endpoint to midpoint difference squared1"
      }
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "second endpoint to midpoint squared distance := sum(625/4, 0)",
      "after": "second endpoint to midpoint squared distance = 625/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依等單位正交L2關係相加兩平方。",
        "en": "Add the two squares in the equal-unit orthogonal L2 frame."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 second endpoint to midpoint squared distance，不以小數近似替代。",
          "en": "Recompute second endpoint to midpoint squared distance with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "625/4",
          "0"
        ],
        "output": "625/4",
        "label": "second endpoint to midpoint squared distance"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "marked vertex to midpoint coordinate difference0 := sub(0, 1/2)",
      "after": "marked vertex to midpoint coordinate difference0 = -1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先取同分量座標差，允許中點有理數。",
        "en": "Subtract matching coordinates,allowing rational midpoint coordinates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 marked vertex to midpoint coordinate difference0，不以小數近似替代。",
          "en": "Recompute marked vertex to midpoint coordinate difference0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "1/2"
        ],
        "output": "-1/2",
        "label": "marked vertex to midpoint coordinate difference0"
      }
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "marked vertex to midpoint difference squared0 := square(-1/2)",
      "after": "marked vertex to midpoint difference squared0 = 1/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐分量平方，差的正負不影響平方長度。",
        "en": "Square each difference;its sign does not affect squared length."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 marked vertex to midpoint difference squared0，不以小數近似替代。",
          "en": "Recompute marked vertex to midpoint difference squared0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "square",
        "inputs": [
          "-1/2"
        ],
        "output": "1/4",
        "label": "marked vertex to midpoint difference squared0"
      }
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "marked vertex to midpoint coordinate difference1 := sub(13, 0)",
      "after": "marked vertex to midpoint coordinate difference1 = 13",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先取同分量座標差，允許中點有理數。",
        "en": "Subtract matching coordinates,allowing rational midpoint coordinates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 marked vertex to midpoint coordinate difference1，不以小數近似替代。",
          "en": "Recompute marked vertex to midpoint coordinate difference1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "13",
          "0"
        ],
        "output": "13",
        "label": "marked vertex to midpoint coordinate difference1"
      }
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "marked vertex to midpoint difference squared1 := square(13)",
      "after": "marked vertex to midpoint difference squared1 = 169",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐分量平方，差的正負不影響平方長度。",
        "en": "Square each difference;its sign does not affect squared length."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 marked vertex to midpoint difference squared1，不以小數近似替代。",
          "en": "Recompute marked vertex to midpoint difference squared1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "square",
        "inputs": [
          "13"
        ],
        "output": "169",
        "label": "marked vertex to midpoint difference squared1"
      }
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "marked vertex to midpoint squared distance := sum(1/4, 169)",
      "after": "marked vertex to midpoint squared distance = 677/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依等單位正交L2關係相加兩平方。",
        "en": "Add the two squares in the equal-unit orthogonal L2 frame."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 marked vertex to midpoint squared distance，不以小數近似替代。",
          "en": "Recompute marked vertex to midpoint squared distance with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/4",
          "169"
        ],
        "output": "677/4",
        "label": "marked vertex to midpoint squared distance"
      }
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "endpoint squares=(625/4,625/4);vertex-to-midpoint square=677/4",
      "after": "endpoint equality does not make every median half the opposite side",
      "relation": "implies",
      "rule_id": "midpoint_endpoint_vs_median_distinction",
      "conditions": [],
      "explanation": {
        "zh-Hant": "中點到兩端點距離相等且各為對邊一半；任意第三頂點到中點的中線長不必等於它們，不把直角斜邊特例套到任意標記。",
        "en": "The midpoint is equally distant from the endpoints,at half the side length.But a general third-vertex median need not have that length;do not apply a right-hypotenuse special case to arbitrary labels."
      },
      "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": [
      "1/2",
      "0",
      "625/4",
      "625/4",
      "677/4"
    ]
  },
  "conclusion": {
    "canonical_value": "(1/2, 0/1, 625/4, 625/4, 677/4)",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261009-advance-20261013-r07-triangle-foundations",
      "batch_path": "public/data/questions/batches/batch-20261009-advance-20261013-r07-triangle-foundations/questions.jsonl",
      "batch_sha256": "5063b1fb565b0934cadf3e54de8bfbf6e9cff8d3ef786ecc721a592ca4db0578",
      "original_line": 88,
      "original_line_sha256": "7e6baac23f16fbd9e71212989dfff53424c18cc8b9cbfd944bebfcab46a61104",
      "question_compact_sha256": "8db0b38c8ac3f2490dd54372354d2cf04cd48f5d61329e06ce589d40f2856400",
      "english_row_compact_sha256": "df64710b9d9edcd0f71e83f0a8685c9086187a610c8b64b0f764319cdbf24a42",
      "english_manifest_path": "public/data/i18n/editions/batch-20261009-advance-20261013-r07-triangle-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261009-advance-20261013-r07-triangle-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-triangle": {
        "sha256": "1556dabd42bf5d132f41296684eb2b037900a511e659c48627bee6d0c4dcd461",
        "english_sha256": "e4a28d8e7440ec3005d4c989132d0cfa5a0c60bf2440ebcd24ca5c2e886f4b5f",
        "original_path": "public/data/mko/triangle.json",
        "english_path": "public/data/i18n/editions/mko-triangle.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "c5c612ca91797c328f4489939ec6f7d885385e63ea21b334c09c421f17933d99",
    "original_candidate_sha256": "f9dc8462b437fb7c884f6ef3d5cea1c26357233a91aa3c7b6fda3e72dce5e3f0",
    "original_candidate_line_sha256": "f9dc8462b437fb7c884f6ef3d5cea1c26357233a91aa3c7b6fda3e72dce5e3f0"
  },
  "historical_source_state": {
    "review_state": {
      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-005-p13",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
    "review_receipts": {
      "semantic_review_sha256": "90dbc494b2d34ef51f412dcff63cfcc1a2215d038bff71f38f5dded4c64faf00",
      "question_first_witness_sha256": "b72a3b8bcf10f584396e0dd5198c657b0934a71864e5a3fa6321d7a613aef9f9"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 31,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "16ea1c3d32d6b4f2e0977e54fd23208ae0d83d82ffe6d4d871c0f7f9efedca08",
      "arithmetic_steps": 26
    },
    "uses_answer_as_premise": false,
    "human_review": "NOT_PERFORMED",
    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
  }
}