Question 3284: 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=(9, 0), A3=(1, 10) and B1=(3, 5), B2=(12, 5), B3=(4, -5), 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=(9, 0), A3=(1, 10) and B1=(3, 5), B2=(12, 5), B3=(4, -5), 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(9, 0)

A vector1 component0 = 9

A vector1 component1 := sub(0, 0)

A vector1 component1 = 0

A vector2 component0 := sub(1, 0)

A vector2 component0 = 1

A vector2 component1 := sub(10, 0)

A vector2 component1 = 10

A determinant positive term := mul(9, 10)

A determinant positive term = 90

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

A determinant negative term = 0

A oriented double area := sub(90, 0)

A oriented double area = 90

determinant A=90

proper triangle=Yes

B vector1 component0 := sub(12, 3)

B vector1 component0 = 9

B vector1 component1 := sub(5, 5)

B vector1 component1 = 0

B vector2 component0 := sub(4, 3)

B vector2 component0 = 1

B vector2 component1 := sub(-5, 5)

B vector2 component1 = -10

B determinant positive term := mul(9, -10)

B determinant positive term = -90

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

B determinant negative term = 0

B oriented double area := sub(-90, 0)

B oriented double area = -90

determinant B=-90

proper triangle=Yes

A12 delta0 := sub(9, 0)

A12 delta0 = 9

A12 square0 := square(9)

A12 square0 = 81

A12 delta1 := sub(0, 0)

A12 delta1 = 0

A12 square1 := square(0)

A12 square1 = 0

A12 squared length := sum(81, 0)

A12 squared length = 81

A13 delta0 := sub(1, 0)

A13 delta0 = 1

A13 square0 := square(1)

A13 square0 = 1

A13 delta1 := sub(10, 0)

A13 delta1 = 10

A13 square1 := square(10)

A13 square1 = 100

A13 squared length := sum(1, 100)

A13 squared length = 101

A23 delta0 := sub(1, 9)

A23 delta0 = -8

A23 square0 := square(-8)

A23 square0 = 64

A23 delta1 := sub(10, 0)

A23 delta1 = 10

A23 square1 := square(10)

A23 square1 = 100

A23 squared length := sum(64, 100)

A23 squared length = 164

B12 delta0 := sub(12, 3)

B12 delta0 = 9

B12 square0 := square(9)

B12 square0 = 81

B12 delta1 := sub(5, 5)

B12 delta1 = 0

B12 square1 := square(0)

B12 square1 = 0

B12 squared length := sum(81, 0)

B12 squared length = 81

B13 delta0 := sub(4, 3)

B13 delta0 = 1

B13 square0 := square(1)

B13 square0 = 1

B13 delta1 := sub(-5, 5)

B13 delta1 = -10

B13 square1 := square(-10)

B13 square1 = 100

B13 squared length := sum(1, 100)

B13 squared length = 101

B23 delta0 := sub(4, 12)

B23 delta0 = -8

B23 square0 := square(-8)

B23 square0 = 64

B23 delta1 := sub(-5, 5)

B23 delta1 = -10

B23 square1 := square(-10)

B23 square1 = 100

B23 squared length := sum(64, 100)

B23 squared length = 164

side 12:A²=81;B²=81

corresponding lengths equal=Yes

side 13:A²=101;B²=101

corresponding lengths equal=Yes

side 23:A²=164;B²=164

corresponding lengths equal=Yes

[Yes, Yes, Yes]

fixed-label SSS holds=Yes

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

Yes

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.