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R0012-06517 · q-20261009-advance-20261013-r07-triangle-0017
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 3]]. 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, 3]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.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.
Before
[[0, 0], [1, 0], [0, 3]]
After
original vertex labels1,2,3 in an equal-unit orthogonal Euclidean frame
Conditions: none
Retain the original A,B,C order;the squared-coordinate formula uses equal units and orthogonal axes.Do not read values from a sketch or assume candidate vertices already form a triangle.
Rule: labelled_triangle_coordinate_scope · Uses: G001
Before
[[0, 0], [1, 0], [0, 3]]
After
three distinct points=Yes
Conditions: none
Check distinctness separately from noncollinearity;collinear points may be distinct,while repeated vertices do not form a proper triangle.
Rule: distinct_triangle_vertex_check · Uses: G001, S001
Before
ABC first vector component0 := sub(1, 0)
After
ABC first vector component0 = 1
Conditions: none
The first edge vector is the second vertex minus the first.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
ABC first vector component1 := sub(0, 0)
After
ABC first vector component1 = 0
Conditions: none
The first edge vector is the second vertex minus the first.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
ABC second vector component0 := sub(0, 0)
After
ABC second vector component0 = 0
Conditions: none
The second edge vector is the third vertex minus that same first vertex.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
ABC second vector component1 := sub(3, 0)
After
ABC second vector component1 = 3
Conditions: none
The second edge vector is the third vertex minus that same first vertex.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
ABC cross positive term := mul(1, 3)
After
ABC cross positive term = 3
Conditions: none
Compute u_x v_y first.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
ABC cross negative term := mul(0, 0)
After
ABC cross negative term = 0
Conditions: none
Compute u_y v_x separately,retaining subtraction for the next step.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
ABC signed determinant := sub(3, 0)
After
ABC signed determinant = 3
Conditions: none
Subtract to obtain signed double area;do not erase label orientation with an absolute value.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
distinct=Yes;signed determinant=3
After
proper triangle=Yes
Conditions: none
A proper triangle needs distinct vertices and nonzero determinant;this judges the supplied coordinates,not a general geometric construction proof.
Rule: triangle_nondegenerate_coordinate_qualification · Uses: G001, S009
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.(Yes, 3, Yes)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 74e8b479da9ff3c38d1fb29aa410d05f41bdc877f9edd4aa8abaf70a22ec623bcandidate record SHA-256: 4ea6360512c997e1a7a52b9b5d48a7f7a0d75b6a4f2fec26f098f415295f4b8asemantic review receipt SHA-256: 90dbc494b2d34ef51f412dcff63cfcc1a2215d038bff71f38f5dded4c64faf00question-first witness SHA-256: b72a3b8bcf10f584396e0dd5198c657b0934a71864e5a3fa6321d7a613aef9f9Lean: NOT_RUN · human review: NOT_PERFORMED