Question 3258: First Mismatch in Fixed Corresponding Sides

Original question and givens

Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(8, 0), A3=(1, 9) and B1=(0, 0), B2=(8, 0), B3=(-1, 9), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. In order 12,13,23, give the first label whose corresponding squared side lengths differ. Answer none if all three agree.

• G001 — Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(8, 0), A3=(1, 9) and B1=(0, 0), B2=(8, 0), B3=(-1, 9), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. In order 12,13,23, give the first label whose corresponding squared side lengths differ. Answer none if all three agree.

Definition and scope

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.

• mko-congruence

Step-by-step derivation

ordered triples in an orthonormal equal-unit frame

vertex indices are fixed;allow rotations,translations and reflections

A vector1 component0 := sub(8, 0)

A vector1 component0 = 8

A vector1 component1 := sub(0, 0)

A vector1 component1 = 0

A vector2 component0 := sub(1, 0)

A vector2 component0 = 1

A vector2 component1 := sub(9, 0)

A vector2 component1 = 9

A determinant positive term := mul(8, 9)

A determinant positive term = 72

A determinant negative term := mul(0, 1)

A determinant negative term = 0

A oriented double area := sub(72, 0)

A oriented double area = 72

determinant A=72

proper triangle=Yes

B vector1 component0 := sub(8, 0)

B vector1 component0 = 8

B vector1 component1 := sub(0, 0)

B vector1 component1 = 0

B vector2 component0 := sub(-1, 0)

B vector2 component0 = -1

B vector2 component1 := sub(9, 0)

B vector2 component1 = 9

B determinant positive term := mul(8, 9)

B determinant positive term = 72

B determinant negative term := mul(0, -1)

B determinant negative term = 0

B oriented double area := sub(72, 0)

B oriented double area = 72

determinant B=72

proper triangle=Yes

A12 delta0 := sub(8, 0)

A12 delta0 = 8

A12 square0 := square(8)

A12 square0 = 64

A12 delta1 := sub(0, 0)

A12 delta1 = 0

A12 square1 := square(0)

A12 square1 = 0

A12 squared length := sum(64, 0)

A12 squared length = 64

A13 delta0 := sub(1, 0)

A13 delta0 = 1

A13 square0 := square(1)

A13 square0 = 1

A13 delta1 := sub(9, 0)

A13 delta1 = 9

A13 square1 := square(9)

A13 square1 = 81

A13 squared length := sum(1, 81)

A13 squared length = 82

A23 delta0 := sub(1, 8)

A23 delta0 = -7

A23 square0 := square(-7)

A23 square0 = 49

A23 delta1 := sub(9, 0)

A23 delta1 = 9

A23 square1 := square(9)

A23 square1 = 81

A23 squared length := sum(49, 81)

A23 squared length = 130

B12 delta0 := sub(8, 0)

B12 delta0 = 8

B12 square0 := square(8)

B12 square0 = 64

B12 delta1 := sub(0, 0)

B12 delta1 = 0

B12 square1 := square(0)

B12 square1 = 0

B12 squared length := sum(64, 0)

B12 squared length = 64

B13 delta0 := sub(-1, 0)

B13 delta0 = -1

B13 square0 := square(-1)

B13 square0 = 1

B13 delta1 := sub(9, 0)

B13 delta1 = 9

B13 square1 := square(9)

B13 square1 = 81

B13 squared length := sum(1, 81)

B13 squared length = 82

B23 delta0 := sub(-1, 8)

B23 delta0 = -9

B23 square0 := square(-9)

B23 square0 = 81

B23 delta1 := sub(9, 0)

B23 delta1 = 9

B23 square1 := square(9)

B23 square1 = 81

B23 squared length := sum(81, 81)

B23 squared length = 162

side 12:A²=64;B²=64

corresponding lengths equal=Yes

side 13:A²=82;B²=82

corresponding lengths equal=Yes

side 23:A²=130;B²=162

corresponding lengths equal=No

[Yes, Yes, No]

fixed-label SSS holds=No

Reverse checks

• S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S050 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S049 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S048 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S047 · premise_and_result_audit — Recompute B23 squared length with exact rational arithmetic, not rounded decimals.
• S046 · premise_and_result_audit — Recompute B23 square1 with exact rational arithmetic, not rounded decimals.
• S045 · premise_and_result_audit — Recompute B23 delta1 with exact rational arithmetic, not rounded decimals.
• S044 · premise_and_result_audit — Recompute B23 square0 with exact rational arithmetic, not rounded decimals.
• S043 · premise_and_result_audit — Recompute B23 delta0 with exact rational arithmetic, not rounded decimals.
• S042 · premise_and_result_audit — Recompute B13 squared length with exact rational arithmetic, not rounded decimals.
• S041 · premise_and_result_audit — Recompute B13 square1 with exact rational arithmetic, not rounded decimals.
• S040 · premise_and_result_audit — Recompute B13 delta1 with exact rational arithmetic, not rounded decimals.
• S039 · premise_and_result_audit — Recompute B13 square0 with exact rational arithmetic, not rounded decimals.
• S038 · premise_and_result_audit — Recompute B13 delta0 with exact rational arithmetic, not rounded decimals.
• S037 · premise_and_result_audit — Recompute B12 squared length with exact rational arithmetic, not rounded decimals.
• S036 · premise_and_result_audit — Recompute B12 square1 with exact rational arithmetic, not rounded decimals.
• S035 · premise_and_result_audit — Recompute B12 delta1 with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Recompute B12 square0 with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Recompute B12 delta0 with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Recompute A23 squared length with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Recompute A23 square1 with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Recompute A23 delta1 with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute A23 square0 with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Recompute A23 delta0 with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute A13 squared length with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Recompute A13 square1 with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute A13 delta1 with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute A13 square0 with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute A13 delta0 with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute A12 squared length with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute A12 square1 with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute A12 delta1 with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute A12 square0 with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute A12 delta0 with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S016 · premise_and_result_audit — Recompute B oriented double area with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute B determinant negative term with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute B determinant positive term with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute B vector2 component1 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute B vector2 component0 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute B vector1 component1 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute B vector1 component0 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Recompute A oriented double area with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute A determinant negative term with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute A determinant positive term with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute A vector2 component1 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute A vector2 component0 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute A vector1 component1 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute A vector1 component0 with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

23

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.