Question 6514: Candidate Labeled Vertex Qualification and Signed Cross

Original question and givens

In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -1]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.

• G001 — In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -1]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.

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-triangle

Step-by-step derivation

[[0, 0], [1, 0], [0, -1]]

original vertex labels1,2,3 in an equal-unit orthogonal Euclidean frame

[[0, 0], [1, 0], [0, -1]]

three distinct points=Yes

ABC first vector component0 := sub(1, 0)

ABC first vector component0 = 1

ABC first vector component1 := sub(0, 0)

ABC first vector component1 = 0

ABC second vector component0 := sub(0, 0)

ABC second vector component0 = 0

ABC second vector component1 := sub(-1, 0)

ABC second vector component1 = -1

ABC cross positive term := mul(1, -1)

ABC cross positive term = -1

ABC cross negative term := mul(0, 0)

ABC cross negative term = 0

ABC signed determinant := sub(-1, 0)

ABC signed determinant = -1

distinct=Yes;signed determinant=-1

proper triangle=Yes

Reverse checks

• S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S009 · premise_and_result_audit — Recompute ABC signed determinant with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute ABC cross negative term with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute ABC cross positive term with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute ABC second vector component1 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute ABC second vector component0 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute ABC first vector component1 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute ABC first vector component0 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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, -1, 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.