第 28880 題：指定正倍率的標記相似見證

原題與已知條件

等單位正交座標系，兩軸採同一長度單位。有序頂點 A=[[0,0],[3,0],[1,2]]、B=[[3,-4],[18,-4],[8,-14]]，兩者非退化且明示對應同索引；指定正長度倍率 k=B/A=5/1。這個倍率是否符合全部三條對應邊？

• G001 — 等單位正交座標系，兩軸採同一長度單位。有序頂點 A=[[0,0],[3,0],[1,2]]、B=[[3,-4],[18,-4],[8,-14]]，兩者非退化且明示對應同索引；指定正長度倍率 k=B/A=5/1。這個倍率是否符合全部三條對應邊？

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-similarity

逐步推導

source roles = [qf-similarity-specified-scale, specified_similarity]

derived = [ALL_fixed_corresponding_sides_at_given_forward_scale]

Similarity ORIGINAL scale numerator squared := mul(5, 5)

Similarity ORIGINAL scale numerator squared = 25

Similarity ORIGINAL scale denominator squared := mul(1, 1)

Similarity ORIGINAL scale denominator squared = 1

Similarity A determinant ORIGINAL point1 coordinate0 := sub(3, 0)

Similarity A determinant ORIGINAL point1 coordinate0 = 3

Similarity A determinant ORIGINAL point1 coordinate1 := sub(0, 0)

Similarity A determinant ORIGINAL point1 coordinate1 = 0

Similarity A determinant ORIGINAL point2 coordinate0 := sub(1, 0)

Similarity A determinant ORIGINAL point2 coordinate0 = 1

Similarity A determinant ORIGINAL point2 coordinate1 := sub(2, 0)

Similarity A determinant ORIGINAL point2 coordinate1 = 2

Similarity A determinant ORIGINAL first cross product := mul(3, 2)

Similarity A determinant ORIGINAL first cross product = 6

Similarity A determinant ORIGINAL second cross product := mul(0, 1)

Similarity A determinant ORIGINAL second cross product = 0

Similarity A determinant ORIGINAL cross difference := sub(6, 0)

Similarity A determinant ORIGINAL cross difference = 6

source roles = [Similarity A, [[0, 0], [3, 0], [1, 2]]]

derived = [6, 是]

Similarity A side12 ORIGINAL coordinate0 difference := sub(3, 0)

Similarity A side12 ORIGINAL coordinate0 difference = 3

Similarity A side12 ORIGINAL coordinate1 difference := sub(0, 0)

Similarity A side12 ORIGINAL coordinate1 difference = 0

Similarity A side12 ORIGINAL coordinate0 square := mul(3, 3)

Similarity A side12 ORIGINAL coordinate0 square = 9

Similarity A side12 ORIGINAL coordinate1 square := mul(0, 0)

Similarity A side12 ORIGINAL coordinate1 square = 0

Similarity A side12 ORIGINAL squared length := sum(9, 0)

Similarity A side12 ORIGINAL squared length = 9

Similarity A side13 ORIGINAL coordinate0 difference := sub(1, 0)

Similarity A side13 ORIGINAL coordinate0 difference = 1

Similarity A side13 ORIGINAL coordinate1 difference := sub(2, 0)

Similarity A side13 ORIGINAL coordinate1 difference = 2

Similarity A side13 ORIGINAL coordinate0 square := mul(1, 1)

Similarity A side13 ORIGINAL coordinate0 square = 1

Similarity A side13 ORIGINAL coordinate1 square := mul(2, 2)

Similarity A side13 ORIGINAL coordinate1 square = 4

Similarity A side13 ORIGINAL squared length := sum(1, 4)

Similarity A side13 ORIGINAL squared length = 5

Similarity A side23 ORIGINAL coordinate0 difference := sub(1, 3)

Similarity A side23 ORIGINAL coordinate0 difference = -2

Similarity A side23 ORIGINAL coordinate1 difference := sub(2, 0)

Similarity A side23 ORIGINAL coordinate1 difference = 2

Similarity A side23 ORIGINAL coordinate0 square := mul(-2, -2)

Similarity A side23 ORIGINAL coordinate0 square = 4

Similarity A side23 ORIGINAL coordinate1 square := mul(2, 2)

Similarity A side23 ORIGINAL coordinate1 square = 4

Similarity A side23 ORIGINAL squared length := sum(4, 4)

Similarity A side23 ORIGINAL squared length = 8

Similarity B determinant ORIGINAL point1 coordinate0 := sub(18, 3)

Similarity B determinant ORIGINAL point1 coordinate0 = 15

Similarity B determinant ORIGINAL point1 coordinate1 := sub(-4, -4)

Similarity B determinant ORIGINAL point1 coordinate1 = 0

Similarity B determinant ORIGINAL point2 coordinate0 := sub(8, 3)

Similarity B determinant ORIGINAL point2 coordinate0 = 5

Similarity B determinant ORIGINAL point2 coordinate1 := sub(-14, -4)

Similarity B determinant ORIGINAL point2 coordinate1 = -10

Similarity B determinant ORIGINAL first cross product := mul(15, -10)

Similarity B determinant ORIGINAL first cross product = -150

Similarity B determinant ORIGINAL second cross product := mul(0, 5)

Similarity B determinant ORIGINAL second cross product = 0

Similarity B determinant ORIGINAL cross difference := sub(-150, 0)

Similarity B determinant ORIGINAL cross difference = -150

source roles = [Similarity B, [[3, -4], [18, -4], [8, -14]]]

derived = [-150, 是]

Similarity B side12 ORIGINAL coordinate0 difference := sub(18, 3)

Similarity B side12 ORIGINAL coordinate0 difference = 15

Similarity B side12 ORIGINAL coordinate1 difference := sub(-4, -4)

Similarity B side12 ORIGINAL coordinate1 difference = 0

Similarity B side12 ORIGINAL coordinate0 square := mul(15, 15)

Similarity B side12 ORIGINAL coordinate0 square = 225

Similarity B side12 ORIGINAL coordinate1 square := mul(0, 0)

Similarity B side12 ORIGINAL coordinate1 square = 0

Similarity B side12 ORIGINAL squared length := sum(225, 0)

Similarity B side12 ORIGINAL squared length = 225

Similarity B side13 ORIGINAL coordinate0 difference := sub(8, 3)

Similarity B side13 ORIGINAL coordinate0 difference = 5

Similarity B side13 ORIGINAL coordinate1 difference := sub(-14, -4)

Similarity B side13 ORIGINAL coordinate1 difference = -10

Similarity B side13 ORIGINAL coordinate0 square := mul(5, 5)

Similarity B side13 ORIGINAL coordinate0 square = 25

Similarity B side13 ORIGINAL coordinate1 square := mul(-10, -10)

Similarity B side13 ORIGINAL coordinate1 square = 100

Similarity B side13 ORIGINAL squared length := sum(25, 100)

Similarity B side13 ORIGINAL squared length = 125

Similarity B side23 ORIGINAL coordinate0 difference := sub(8, 18)

Similarity B side23 ORIGINAL coordinate0 difference = -10

Similarity B side23 ORIGINAL coordinate1 difference := sub(-14, -4)

Similarity B side23 ORIGINAL coordinate1 difference = -10

Similarity B side23 ORIGINAL coordinate0 square := mul(-10, -10)

Similarity B side23 ORIGINAL coordinate0 square = 100

Similarity B side23 ORIGINAL coordinate1 square := mul(-10, -10)

Similarity B side23 ORIGINAL coordinate1 square = 100

Similarity B side23 ORIGINAL squared length := sum(100, 100)

Similarity B side23 ORIGINAL squared length = 200

Similarity fixed side0 ORIGINAL Bsquare times q squared := mul(225, 1)

Similarity fixed side0 ORIGINAL Bsquare times q squared = 225

Similarity fixed side0 ORIGINAL Asquare times p squared := mul(9, 25)

Similarity fixed side0 ORIGINAL Asquare times p squared = 225

source roles = [0, 9, 225, 5, 1]

derived = [225, 225, 是]

Similarity fixed side1 ORIGINAL Bsquare times q squared := mul(125, 1)

Similarity fixed side1 ORIGINAL Bsquare times q squared = 125

Similarity fixed side1 ORIGINAL Asquare times p squared := mul(5, 25)

Similarity fixed side1 ORIGINAL Asquare times p squared = 125

source roles = [1, 5, 125, 5, 1]

derived = [125, 125, 是]

Similarity fixed side2 ORIGINAL Bsquare times q squared := mul(200, 1)

Similarity fixed side2 ORIGINAL Bsquare times q squared = 200

Similarity fixed side2 ORIGINAL Asquare times p squared := mul(8, 25)

Similarity fixed side2 ORIGINAL Asquare times p squared = 200

source roles = [2, 8, 200, 5, 1]

derived = [200, 200, 是]

source roles = [[[0, 0], [3, 0], [1, 2]], [[3, -4], [18, -4], [8, -14]], 5, 1]

derived = [[是, 是, 是], 是, 是]

逐步反向核對

• S059 · premise_and_result_audit — 反向逐項代回原始資料與本步條件；保持索引、符號、嚴格／非嚴格邊界，不能用正確答案代替推導責任。這是候選教學與精確核對，非形式證明或發布資格。
• S058 · premise_and_result_audit — 反向逐項代回原始資料與本步條件；保持索引、符號、嚴格／非嚴格邊界，不能用正確答案代替推導責任。這是候選教學與精確核對，非形式證明或發布資格。
• S057 · premise_and_result_audit — 以精確有理數重算 Similarity fixed side2 ORIGINAL Asquare times p squared，不以小數近似替代。
• S056 · premise_and_result_audit — 以精確有理數重算 Similarity fixed side2 ORIGINAL Bsquare times q squared，不以小數近似替代。
• S055 · premise_and_result_audit — 反向逐項代回原始資料與本步條件；保持索引、符號、嚴格／非嚴格邊界，不能用正確答案代替推導責任。這是候選教學與精確核對，非形式證明或發布資格。
• S054 · premise_and_result_audit — 以精確有理數重算 Similarity fixed side1 ORIGINAL Asquare times p squared，不以小數近似替代。
• S053 · premise_and_result_audit — 以精確有理數重算 Similarity fixed side1 ORIGINAL Bsquare times q squared，不以小數近似替代。
• S052 · premise_and_result_audit — 反向逐項代回原始資料與本步條件；保持索引、符號、嚴格／非嚴格邊界，不能用正確答案代替推導責任。這是候選教學與精確核對，非形式證明或發布資格。
• S051 · premise_and_result_audit — 以精確有理數重算 Similarity fixed side0 ORIGINAL Asquare times p squared，不以小數近似替代。
• S050 · premise_and_result_audit — 以精確有理數重算 Similarity fixed side0 ORIGINAL Bsquare times q squared，不以小數近似替代。
• S049 · premise_and_result_audit — 以精確有理數重算 Similarity B side23 ORIGINAL squared length，不以小數近似替代。
• S048 · premise_and_result_audit — 以精確有理數重算 Similarity B side23 ORIGINAL coordinate1 square，不以小數近似替代。
• S047 · premise_and_result_audit — 以精確有理數重算 Similarity B side23 ORIGINAL coordinate0 square，不以小數近似替代。
• S046 · premise_and_result_audit — 以精確有理數重算 Similarity B side23 ORIGINAL coordinate1 difference，不以小數近似替代。
• S045 · premise_and_result_audit — 以精確有理數重算 Similarity B side23 ORIGINAL coordinate0 difference，不以小數近似替代。
• S044 · premise_and_result_audit — 以精確有理數重算 Similarity B side13 ORIGINAL squared length，不以小數近似替代。
• S043 · premise_and_result_audit — 以精確有理數重算 Similarity B side13 ORIGINAL coordinate1 square，不以小數近似替代。
• S042 · premise_and_result_audit — 以精確有理數重算 Similarity B side13 ORIGINAL coordinate0 square，不以小數近似替代。
• S041 · premise_and_result_audit — 以精確有理數重算 Similarity B side13 ORIGINAL coordinate1 difference，不以小數近似替代。
• S040 · premise_and_result_audit — 以精確有理數重算 Similarity B side13 ORIGINAL coordinate0 difference，不以小數近似替代。
• S039 · premise_and_result_audit — 以精確有理數重算 Similarity B side12 ORIGINAL squared length，不以小數近似替代。
• S038 · premise_and_result_audit — 以精確有理數重算 Similarity B side12 ORIGINAL coordinate1 square，不以小數近似替代。
• S037 · premise_and_result_audit — 以精確有理數重算 Similarity B side12 ORIGINAL coordinate0 square，不以小數近似替代。
• S036 · premise_and_result_audit — 以精確有理數重算 Similarity B side12 ORIGINAL coordinate1 difference，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 Similarity B side12 ORIGINAL coordinate0 difference，不以小數近似替代。
• S034 · premise_and_result_audit — 反向逐項代回原始資料與本步條件；保持索引、符號、嚴格／非嚴格邊界，不能用正確答案代替推導責任。這是候選教學與精確核對，非形式證明或發布資格。
• S033 · premise_and_result_audit — 以精確有理數重算 Similarity B determinant ORIGINAL cross difference，不以小數近似替代。
• S032 · premise_and_result_audit — 以精確有理數重算 Similarity B determinant ORIGINAL second cross product，不以小數近似替代。
• S031 · premise_and_result_audit — 以精確有理數重算 Similarity B determinant ORIGINAL first cross product，不以小數近似替代。
• S030 · premise_and_result_audit — 以精確有理數重算 Similarity B determinant ORIGINAL point2 coordinate1，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 Similarity B determinant ORIGINAL point2 coordinate0，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 Similarity B determinant ORIGINAL point1 coordinate1，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 Similarity B determinant ORIGINAL point1 coordinate0，不以小數近似替代。
• S026 · premise_and_result_audit — 以精確有理數重算 Similarity A side23 ORIGINAL squared length，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 Similarity A side23 ORIGINAL coordinate1 square，不以小數近似替代。
• S024 · premise_and_result_audit — 以精確有理數重算 Similarity A side23 ORIGINAL coordinate0 square，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 Similarity A side23 ORIGINAL coordinate1 difference，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 Similarity A side23 ORIGINAL coordinate0 difference，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 Similarity A side13 ORIGINAL squared length，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 Similarity A side13 ORIGINAL coordinate1 square，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 Similarity A side13 ORIGINAL coordinate0 square，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 Similarity A side13 ORIGINAL coordinate1 difference，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 Similarity A side13 ORIGINAL coordinate0 difference，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 Similarity A side12 ORIGINAL squared length，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 Similarity A side12 ORIGINAL coordinate1 square，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 Similarity A side12 ORIGINAL coordinate0 square，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 Similarity A side12 ORIGINAL coordinate1 difference，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 Similarity A side12 ORIGINAL coordinate0 difference，不以小數近似替代。
• S011 · premise_and_result_audit — 反向逐項代回原始資料與本步條件；保持索引、符號、嚴格／非嚴格邊界，不能用正確答案代替推導責任。這是候選教學與精確核對，非形式證明或發布資格。
• S010 · premise_and_result_audit — 以精確有理數重算 Similarity A determinant ORIGINAL cross difference，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 Similarity A determinant ORIGINAL second cross product，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 Similarity A determinant ORIGINAL first cross product，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 Similarity A determinant ORIGINAL point2 coordinate1，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 Similarity A determinant ORIGINAL point2 coordinate0，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 Similarity A determinant ORIGINAL point1 coordinate1，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 Similarity A determinant ORIGINAL point1 coordinate0，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 Similarity ORIGINAL scale denominator squared，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 Similarity ORIGINAL scale numerator squared，不以小數近似替代。
• S001 · premise_and_result_audit — 反向逐項代回原始資料與本步條件；保持索引、符號、嚴格／非嚴格邊界，不能用正確答案代替推導責任。這是候選教學與精確核對，非形式證明或發布資格。

結論

是

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。