# Question 3315: Triangle Eligibility of an Ordered Point Triple [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-0090) 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. R0005-03315 · q-20261006-advance-20261007-r05-congruence-0090 ## Original question and givens Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered points are A1=(0, 0), A2=(14, 0), A3=(14, 0). Do they form a nondegenerate triangle of distinct noncollinear points? Reject repeated or collinear data. - G001 — Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered points are A1=(0, 0), A2=(14, 0), A3=(14, 0). Do they form a nondegenerate triangle of distinct noncollinear points? Reject repeated or collinear data. ## 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(14, 0) After A vector1 component0 = 14 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(14, 0) After A vector2 component0 = 14 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(0, 0) After A vector2 component1 = 0 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(14, 0) After A determinant positive term = 0 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, 14) 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(0, 0) After A oriented double area = 0 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=0 After proper triangle=No 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 ## Reverse checks - 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: 54a824b316cd2c261d20a7aa2ae1b8d8ec3a378e12c6d12914498ad18dc9c158 - candidate record SHA-256: 7630e9fcdb72327415e375e4c6aac6395edd2957584fa36650c81e968c96572f - 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-0090.json)