Question 3308: Triangle Eligibility of an Ordered Point Triple

Original question and givens

Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered points are A1=(0, 0), A2=(7, 0), A3=(8, 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=(7, 0), A3=(8, 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

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(7, 0)

A vector1 component0 = 7

A vector1 component1 := sub(0, 0)

A vector1 component1 = 0

A vector2 component0 := sub(8, 0)

A vector2 component0 = 8

A vector2 component1 := sub(0, 0)

A vector2 component1 = 0

A determinant positive term := mul(7, 0)

A determinant positive term = 0

A determinant negative term := mul(0, 8)

A determinant negative term = 0

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

A oriented double area = 0

determinant A=0

proper triangle=No

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.