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R0012-06576 · q-20261009-advance-20261013-r07-triangle-0026
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 1], [0, 0], [1, 0]]. The triangle is qualified as distinct and noncollinear, vertex=1. Give (other two vertex indices, opposite side’s squared L2 distance), not its distance.
G001 — In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 1], [0, 0], [1, 0]]. The triangle is qualified as distinct and noncollinear, vertex=1. Give (other two vertex indices, opposite side’s squared L2 distance), not its distance.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, 1], [0, 0], [1, 0]]
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, 1], [0, 0], [1, 0]]
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(0, 0)
After
ABC first vector component0 = 0
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, 1)
After
ABC first vector component1 = -1
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(1, 0)
After
ABC second vector component0 = 1
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(0, 1)
After
ABC second vector component1 = -1
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(0, -1)
After
ABC cross positive term = 0
Conditions: none
Compute u_x v_y first.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
ABC cross negative term := mul(-1, 1)
After
ABC cross negative term = -1
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(0, -1)
After
ABC signed determinant = 1
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=1
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
Before
marked vertex=1
After
opposite endpoints=[2, 3]
Conditions: none
The opposite-side endpoints are the other two original labels,not a reordered coordinate or side-length list.
Rule: original_label_opposite_side · Uses: G001, S010
Before
opposite side coordinate difference0 := sub(0, 1)
After
opposite side coordinate difference0 = -1
Conditions: none
Subtract matching coordinates,allowing rational midpoint coordinates.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
opposite side difference squared0 := square(-1)
After
opposite side difference squared0 = 1
Conditions: none
Square each difference;its sign does not affect squared length.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
opposite side coordinate difference1 := sub(0, 0)
After
opposite side coordinate difference1 = 0
Conditions: none
Subtract matching coordinates,allowing rational midpoint coordinates.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
opposite side difference squared1 := square(0)
After
opposite side difference squared1 = 0
Conditions: none
Square each difference;its sign does not affect squared length.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
opposite side squared distance := sum(1, 0)
After
opposite side squared distance = 1
Conditions: none
Add the two squares in the equal-unit orthogonal L2 frame.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
([2, 3], 1)
After
second field is squared length,not length
Conditions: none
Do not call squared distance the side length or reorder endpoint labels because of the chosen vertex.
Rule: opposite_side_square_not_length · Uses: G001, S016
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 opposite side squared distance with exact rational arithmetic, not rounded decimals.S015 · premise_and_result_audit — Recompute opposite side difference squared1 with exact rational arithmetic, not rounded decimals.S014 · premise_and_result_audit — Recompute opposite side coordinate difference1 with exact rational arithmetic, not rounded decimals.S013 · premise_and_result_audit — Recompute opposite side difference squared0 with exact rational arithmetic, not rounded decimals.S012 · premise_and_result_audit — Recompute opposite side coordinate difference0 with exact rational arithmetic, not rounded decimals.S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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.([2, 3], 1)
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.
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