# Question 3287: SSS Congruence with Fixed Vertex Labels [Return to catalogue page 17](https://ocme.evemisslab.com/en/questions/page/17/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261006-advance-20261007-r05-congruence-foundations/#q-20261006-advance-20261007-r05-congruence-0012) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0005-03287 · q-20261006-advance-20261007-r05-congruence-0012 ## Original question and givens Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(12, 0), A3=(1, 13) and B1=(0, 0), B2=(12, 0), B3=(-1, 13), 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=(12, 0), A3=(1, 13) and B1=(0, 0), B2=(12, 0), B3=(-1, 13), 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](https://ocme.evemisslab.com/en/mko/mko-congruence/) ## Step-by-step derivation S001 · impliesBefore ordered triples in an orthonormal equal-unit frame After vertex indices are fixed;allow rotations,translations and reflections Conditions: none Correspond equal indices without silently permuting vertices;Euclidean congruence allows reflection.Check nondegeneracy before SSS. Step identifiers and dependenciesRule: fixed_vertex_correspondence · Uses: G001 S002 · calculationBefore A vector1 component0 := sub(12, 0) After A vector1 component0 = 12 Conditions: none Subtract the first vertex to obtain an edge-vector coordinate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore A vector1 component1 := sub(0, 0) After A vector1 component1 = 0 Conditions: none Subtract the first vertex to obtain an edge-vector coordinate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore A vector2 component0 := sub(1, 0) After A vector2 component0 = 1 Conditions: none Subtract the first vertex to obtain an edge-vector coordinate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore A vector2 component1 := sub(13, 0) After A vector2 component1 = 13 Conditions: none Subtract the first vertex to obtain an edge-vector coordinate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore A determinant positive term := mul(12, 13) After A determinant positive term = 156 Conditions: none Compute the first cross product term of the determinant. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore A determinant negative term := mul(0, 1) After A determinant negative term = 0 Conditions: none Compute the other cross product term,retaining its subtraction position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore A oriented double area := sub(156, 0) After A oriented double area = 156 Conditions: none Subtract the terms;nonzero means noncollinear and also excludes repeated vertices. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · impliesBefore determinant A=156 After proper triangle=Yes Conditions: none This exact determinant replaces visual guessing;degenerate triples do not meet the triangle premise of SSS. Step identifiers and dependenciesRule: triangle_nondegeneracy · Uses: G001, S008 S010 · calculationBefore B vector1 component0 := sub(12, 0) After B vector1 component0 = 12 Conditions: none Subtract the first vertex to obtain an edge-vector coordinate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore B vector1 component1 := sub(0, 0) After B vector1 component1 = 0 Conditions: none Subtract the first vertex to obtain an edge-vector coordinate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore B vector2 component0 := sub(-1, 0) After B vector2 component0 = -1 Conditions: none Subtract the first vertex to obtain an edge-vector coordinate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore B vector2 component1 := sub(13, 0) After B vector2 component1 = 13 Conditions: none Subtract the first vertex to obtain an edge-vector coordinate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore B determinant positive term := mul(12, 13) After B determinant positive term = 156 Conditions: none Compute the first cross product term of the determinant. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore B determinant negative term := mul(0, -1) After B determinant negative term = 0 Conditions: none Compute the other cross product term,retaining its subtraction position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore B oriented double area := sub(156, 0) After B oriented double area = 156 Conditions: none Subtract the terms;nonzero means noncollinear and also excludes repeated vertices. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · impliesBefore determinant B=156 After proper triangle=Yes Conditions: none This exact determinant replaces visual guessing;degenerate triples do not meet the triangle premise of SSS. Step identifiers and dependenciesRule: triangle_nondegeneracy · Uses: G001, S016 S018 · calculationBefore A12 delta0 := sub(12, 0) After A12 delta0 = 12 Conditions: none Subtract coordinates of the endpoints of this same side. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore A12 square0 := square(12) After A12 square0 = 144 Conditions: none Square the coordinate difference;its sign does not affect length. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore A12 delta1 := sub(0, 0) After A12 delta1 = 0 Conditions: none Subtract coordinates of the endpoints of this same side. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore A12 square1 := square(0) After A12 square1 = 0 Conditions: none Square the coordinate difference;its sign does not affect length. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore A12 squared length := sum(144, 0) After A12 squared length = 144 Conditions: none Add by the Pythagorean relation to obtain squared length,not length itself. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore A13 delta0 := sub(1, 0) After A13 delta0 = 1 Conditions: none Subtract coordinates of the endpoints of this same side. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore A13 square0 := square(1) After A13 square0 = 1 Conditions: none Square the coordinate difference;its sign does not affect length. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore A13 delta1 := sub(13, 0) After A13 delta1 = 13 Conditions: none Subtract coordinates of the endpoints of this same side. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · calculationBefore A13 square1 := square(13) After A13 square1 = 169 Conditions: none Square the coordinate difference;its sign does not affect length. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · calculationBefore A13 squared length := sum(1, 169) After A13 squared length = 170 Conditions: none Add by the Pythagorean relation to obtain squared length,not length itself. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · calculationBefore A23 delta0 := sub(1, 12) After A23 delta0 = -11 Conditions: none Subtract coordinates of the endpoints of this same side. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · calculationBefore A23 square0 := square(-11) After A23 square0 = 121 Conditions: none Square the coordinate difference;its sign does not affect length. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028 S030 · calculationBefore A23 delta1 := sub(13, 0) After A23 delta1 = 13 Conditions: none Subtract coordinates of the endpoints of this same side. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S029 S031 · calculationBefore A23 square1 := square(13) After A23 square1 = 169 Conditions: none Square the coordinate difference;its sign does not affect length. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030 S032 · calculationBefore A23 squared length := sum(121, 169) After A23 squared length = 290 Conditions: none Add by the Pythagorean relation to obtain squared length,not length itself. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031 S033 · calculationBefore B12 delta0 := sub(12, 0) After B12 delta0 = 12 Conditions: none Subtract coordinates of the endpoints of this same side. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032 S034 · calculationBefore B12 square0 := square(12) After B12 square0 = 144 Conditions: none Square the coordinate difference;its sign does not affect length. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S033 S035 · calculationBefore B12 delta1 := sub(0, 0) After B12 delta1 = 0 Conditions: none Subtract coordinates of the endpoints of this same side. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034 S036 · calculationBefore B12 square1 := square(0) After B12 square1 = 0 Conditions: none Square the coordinate difference;its sign does not affect length. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S035 S037 · calculationBefore B12 squared length := sum(144, 0) After B12 squared length = 144 Conditions: none Add by the Pythagorean relation to obtain squared length,not length itself. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S036 S038 · calculationBefore B13 delta0 := sub(-1, 0) After B13 delta0 = -1 Conditions: none Subtract coordinates of the endpoints of this same side. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S037 S039 · calculationBefore B13 square0 := square(-1) After B13 square0 = 1 Conditions: none Square the coordinate difference;its sign does not affect length. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S038 S040 · calculationBefore B13 delta1 := sub(13, 0) After B13 delta1 = 13 Conditions: none Subtract coordinates of the endpoints of this same side. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S039 S041 · calculationBefore B13 square1 := square(13) After B13 square1 = 169 Conditions: none Square the coordinate difference;its sign does not affect length. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S040 S042 · calculationBefore B13 squared length := sum(1, 169) After B13 squared length = 170 Conditions: none Add by the Pythagorean relation to obtain squared length,not length itself. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S041 S043 · calculationBefore B23 delta0 := sub(-1, 12) After B23 delta0 = -13 Conditions: none Subtract coordinates of the endpoints of this same side. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S042 S044 · calculationBefore B23 square0 := square(-13) After B23 square0 = 169 Conditions: none Square the coordinate difference;its sign does not affect length. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S043 S045 · calculationBefore B23 delta1 := sub(13, 0) After B23 delta1 = 13 Conditions: none Subtract coordinates of the endpoints of this same side. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S044 S046 · calculationBefore B23 square1 := square(13) After B23 square1 = 169 Conditions: none Square the coordinate difference;its sign does not affect length. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S045 S047 · calculationBefore B23 squared length := sum(169, 169) After B23 squared length = 338 Conditions: none Add by the Pythagorean relation to obtain squared length,not length itself. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S046 S048 · impliesBefore side 12:A²=144;B²=144 After corresponding lengths equal=Yes Conditions: none Lengths are nonnegative,so equality of squares is equivalent to equality of lengths.Compare only sides with the same fixed labels. Step identifiers and dependenciesRule: corresponding_side_equality · Uses: G001, S047 S049 · impliesBefore side 13:A²=170;B²=170 After corresponding lengths equal=Yes Conditions: none Lengths are nonnegative,so equality of squares is equivalent to equality of lengths.Compare only sides with the same fixed labels. Step identifiers and dependenciesRule: corresponding_side_equality · Uses: G001, S048 S050 · impliesBefore side 23:A²=290;B²=338 After corresponding lengths equal=No Conditions: none Lengths are nonnegative,so equality of squares is equivalent to equality of lengths.Compare only sides with the same fixed labels. Step identifiers and dependenciesRule: corresponding_side_equality · Uses: G001, S049 S051 · impliesBefore [Yes, Yes, No] After fixed-label SSS holds=No Conditions: none SSS requires all three corresponding sides of the proper triangles to agree.The three sides determine the inner products of two edge vectors,allowing a Euclidean isometry.Any mismatch rejects this labeling,not every other permutation. Step identifiers and dependenciesRule: fixed_label_sss · Uses: G001, S050 ## 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: d665b6e7590bfe4da16b82a267df57dd8945eec3aee06fa7ecf215e1ae2c87c5 - candidate record SHA-256: 3a9b3abacddaa37d43d10cd2c03c01b480957b65e9708b4f022b6a23c796e82d - semantic review receipt SHA-256: 5d74de11610ebc6037a4fc9946d77154d4ed18389f8f0feeb823148b6320c4c2 - question-first witness SHA-256: 1fba4ab0f13c2cfcc4d9fdebe492c442d43dd45700394a443daa28f333355ed9 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261006-advance-20261007-r05-congruence-0012.json)