Question 3295: SSS Congruence with Fixed Vertex Labels

Original question and givens

Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(20, 0), A3=(1, 21) and B1=(0, 0), B2=(20, 0), B3=(-1, 21), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Do all three corresponding sides satisfy labeled SSS congruence?

• G001 — Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(20, 0), A3=(1, 21) and B1=(0, 0), B2=(20, 0), B3=(-1, 21), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. Do all three corresponding sides satisfy labeled SSS congruence?

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(20, 0)

A vector1 component0 = 20

A vector1 component1 := sub(0, 0)

A vector1 component1 = 0

A vector2 component0 := sub(1, 0)

A vector2 component0 = 1

A vector2 component1 := sub(21, 0)

A vector2 component1 = 21

A determinant positive term := mul(20, 21)

A determinant positive term = 420

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

A determinant negative term = 0

A oriented double area := sub(420, 0)

A oriented double area = 420

determinant A=420

proper triangle=Yes

B vector1 component0 := sub(20, 0)

B vector1 component0 = 20

B vector1 component1 := sub(0, 0)

B vector1 component1 = 0

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

B vector2 component0 = -1

B vector2 component1 := sub(21, 0)

B vector2 component1 = 21

B determinant positive term := mul(20, 21)

B determinant positive term = 420

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

B determinant negative term = 0

B oriented double area := sub(420, 0)

B oriented double area = 420

determinant B=420

proper triangle=Yes

A12 delta0 := sub(20, 0)

A12 delta0 = 20

A12 square0 := square(20)

A12 square0 = 400

A12 delta1 := sub(0, 0)

A12 delta1 = 0

A12 square1 := square(0)

A12 square1 = 0

A12 squared length := sum(400, 0)

A12 squared length = 400

A13 delta0 := sub(1, 0)

A13 delta0 = 1

A13 square0 := square(1)

A13 square0 = 1

A13 delta1 := sub(21, 0)

A13 delta1 = 21

A13 square1 := square(21)

A13 square1 = 441

A13 squared length := sum(1, 441)

A13 squared length = 442

A23 delta0 := sub(1, 20)

A23 delta0 = -19

A23 square0 := square(-19)

A23 square0 = 361

A23 delta1 := sub(21, 0)

A23 delta1 = 21

A23 square1 := square(21)

A23 square1 = 441

A23 squared length := sum(361, 441)

A23 squared length = 802

B12 delta0 := sub(20, 0)

B12 delta0 = 20

B12 square0 := square(20)

B12 square0 = 400

B12 delta1 := sub(0, 0)

B12 delta1 = 0

B12 square1 := square(0)

B12 square1 = 0

B12 squared length := sum(400, 0)

B12 squared length = 400

B13 delta0 := sub(-1, 0)

B13 delta0 = -1

B13 square0 := square(-1)

B13 square0 = 1

B13 delta1 := sub(21, 0)

B13 delta1 = 21

B13 square1 := square(21)

B13 square1 = 441

B13 squared length := sum(1, 441)

B13 squared length = 442

B23 delta0 := sub(-1, 20)

B23 delta0 = -21

B23 square0 := square(-21)

B23 square0 = 441

B23 delta1 := sub(21, 0)

B23 delta1 = 21

B23 square1 := square(21)

B23 square1 = 441

B23 squared length := sum(441, 441)

B23 squared length = 882

side 12:A²=400;B²=400

corresponding lengths equal=Yes

side 13:A²=442;B²=442

corresponding lengths equal=Yes

side 23:A²=802;B²=882

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

No

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.