第 6563 題：對邊中點與三段平方

原題與已知條件

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。

• G001 — 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。

定義與適用範圍

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。

• mko-triangle

逐步推導

[[0, 13], [-12, 0], [13, 0]]

original vertex labels1,2,3 in an equal-unit orthogonal Euclidean frame

[[0, 13], [-12, 0], [13, 0]]

three distinct points=是

ABC first vector component0 := sub(-12, 0)

ABC first vector component0 = -12

ABC first vector component1 := sub(0, 13)

ABC first vector component1 = -13

ABC second vector component0 := sub(13, 0)

ABC second vector component0 = 13

ABC second vector component1 := sub(0, 13)

ABC second vector component1 = -13

ABC cross positive term := mul(-12, -13)

ABC cross positive term = 156

ABC cross negative term := mul(-13, 13)

ABC cross negative term = -169

ABC signed determinant := sub(156, -169)

ABC signed determinant = 325

distinct=是;signed determinant=325

proper triangle=是

marked vertex=1

opposite endpoints=[2, 3]

midpoint coordinate sum0 := sum(-12, 13)

midpoint coordinate sum0 = 1

midpoint coordinate0 := div(1, 2)

midpoint coordinate0 = 1/2

midpoint coordinate sum1 := sum(0, 0)

midpoint coordinate sum1 = 0

midpoint coordinate1 := div(0, 2)

midpoint coordinate1 = 0

first endpoint to midpoint coordinate difference0 := sub(-12, 1/2)

first endpoint to midpoint coordinate difference0 = -25/2

first endpoint to midpoint difference squared0 := square(-25/2)

first endpoint to midpoint difference squared0 = 625/4

first endpoint to midpoint coordinate difference1 := sub(0, 0)

first endpoint to midpoint coordinate difference1 = 0

first endpoint to midpoint difference squared1 := square(0)

first endpoint to midpoint difference squared1 = 0

first endpoint to midpoint squared distance := sum(625/4, 0)

first endpoint to midpoint squared distance = 625/4

second endpoint to midpoint coordinate difference0 := sub(13, 1/2)

second endpoint to midpoint coordinate difference0 = 25/2

second endpoint to midpoint difference squared0 := square(25/2)

second endpoint to midpoint difference squared0 = 625/4

second endpoint to midpoint coordinate difference1 := sub(0, 0)

second endpoint to midpoint coordinate difference1 = 0

second endpoint to midpoint difference squared1 := square(0)

second endpoint to midpoint difference squared1 = 0

second endpoint to midpoint squared distance := sum(625/4, 0)

second endpoint to midpoint squared distance = 625/4

marked vertex to midpoint coordinate difference0 := sub(0, 1/2)

marked vertex to midpoint coordinate difference0 = -1/2

marked vertex to midpoint difference squared0 := square(-1/2)

marked vertex to midpoint difference squared0 = 1/4

marked vertex to midpoint coordinate difference1 := sub(13, 0)

marked vertex to midpoint coordinate difference1 = 13

marked vertex to midpoint difference squared1 := square(13)

marked vertex to midpoint difference squared1 = 169

marked vertex to midpoint squared distance := sum(1/4, 169)

marked vertex to midpoint squared distance = 677/4

endpoint squares=(625/4,625/4);vertex-to-midpoint square=677/4

endpoint equality does not make every median half the opposite side

逐步反向核對

• S031 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S030 · premise_and_result_audit — 以精確有理數重算 marked vertex to midpoint squared distance，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 marked vertex to midpoint difference squared1，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 marked vertex to midpoint coordinate difference1，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 marked vertex to midpoint difference squared0，不以小數近似替代。
• S026 · premise_and_result_audit — 以精確有理數重算 marked vertex to midpoint coordinate difference0，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 second endpoint to midpoint squared distance，不以小數近似替代。
• S024 · premise_and_result_audit — 以精確有理數重算 second endpoint to midpoint difference squared1，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 second endpoint to midpoint coordinate difference1，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 second endpoint to midpoint difference squared0，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 second endpoint to midpoint coordinate difference0，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 first endpoint to midpoint squared distance，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 first endpoint to midpoint difference squared1，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 first endpoint to midpoint coordinate difference1，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 first endpoint to midpoint difference squared0，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 first endpoint to midpoint coordinate difference0，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 midpoint coordinate1，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 midpoint coordinate sum1，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 midpoint coordinate0，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 midpoint coordinate sum0，不以小數近似替代。
• S011 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S010 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S009 · premise_and_result_audit — 以精確有理數重算 ABC signed determinant，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 ABC cross negative term，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 ABC cross positive term，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 ABC second vector component1，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 ABC second vector component0，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 ABC first vector component1，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 ABC first vector component0，不以小數近似替代。
• S002 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。

結論

(1/2, 0/1, 625/4, 625/4, 677/4)

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。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。