Labeled Euclidean Vertex and Midpoint Checks
This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.
This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261009-advance-20261013-r07-triangle-foundations · 2026-10-09 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Candidate Labeled Vertex Qualification and Signed Cross
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [0, 0], [1, 1]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.
View answer and explanation
(No, 0, No)
Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (No, 0, No).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-coordinate-qualification · candidate
2. Candidate Labeled Vertex Qualification and Signed Cross
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 1], [2, 2]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.
View answer and explanation
(Yes, 0, No)
Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, 0, No).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-coordinate-qualification · candidate
3. Candidate Labeled Vertex Qualification and Signed Cross
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -12]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.
View answer and explanation
(Yes, -12, Yes)
Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, -12, Yes).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-coordinate-qualification · candidate
4. Candidate Labeled Vertex Qualification and Signed Cross
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -11]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.
View answer and explanation
(Yes, -11, Yes)
Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, -11, Yes).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-coordinate-qualification · candidate
5. Candidate Labeled Vertex Qualification and Signed Cross
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -10]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.
View answer and explanation
(Yes, -10, Yes)
Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, -10, Yes).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-coordinate-qualification · candidate
6. Candidate Labeled Vertex Qualification and Signed Cross
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -9]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.
View answer and explanation
(Yes, -9, Yes)
Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, -9, Yes).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-coordinate-qualification · candidate
7. Candidate Labeled Vertex Qualification and Signed Cross
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -8]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.
View answer and explanation
(Yes, -8, Yes)
Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, -8, Yes).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-coordinate-qualification · candidate
8. Candidate Labeled Vertex Qualification and Signed Cross
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -7]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.
View answer and explanation
(Yes, -7, Yes)
Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, -7, Yes).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-coordinate-qualification · candidate
9. Candidate Labeled Vertex Qualification and Signed Cross
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -6]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.
View answer and explanation
(Yes, -6, Yes)
Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, -6, Yes).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-coordinate-qualification · candidate
10. Candidate Labeled Vertex Qualification and Signed Cross
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -5]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.
View answer and explanation
(Yes, -5, Yes)
Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, -5, Yes).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-coordinate-qualification · candidate
11. Candidate Labeled Vertex Qualification and Signed Cross
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -4]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.
View answer and explanation
(Yes, -4, Yes)
Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, -4, Yes).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-coordinate-qualification · candidate
12. Candidate Labeled Vertex Qualification and Signed Cross
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.
View answer and explanation
(Yes, -3, Yes)
Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, -3, Yes).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-coordinate-qualification · candidate
13. Candidate Labeled Vertex Qualification and Signed Cross
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -2]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.
View answer and explanation
(Yes, -2, Yes)
Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, -2, Yes).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-coordinate-qualification · candidate
14. Candidate Labeled Vertex Qualification and Signed Cross
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.
View answer and explanation
(Yes, -1, Yes)
Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, -1, Yes).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-coordinate-qualification · candidate
15. Candidate Labeled Vertex Qualification and Signed Cross
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.
View answer and explanation
(Yes, 1, Yes)
Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, 1, Yes).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-coordinate-qualification · candidate
16. Candidate Labeled Vertex Qualification and Signed Cross
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 2]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.
View answer and explanation
(Yes, 2, Yes)
Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, 2, Yes).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-coordinate-qualification · candidate
17. Candidate Labeled Vertex Qualification and Signed Cross
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.
View answer and explanation
(Yes, 3, Yes)
Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, 3, Yes).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-coordinate-qualification · candidate
18. Candidate Labeled Vertex Qualification and Signed Cross
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 4]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.
View answer and explanation
(Yes, 4, Yes)
Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, 4, Yes).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-coordinate-qualification · candidate
19. Candidate Labeled Vertex Qualification and Signed Cross
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 5]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.
View answer and explanation
(Yes, 5, Yes)
Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, 5, Yes).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-coordinate-qualification · candidate
20. Candidate Labeled Vertex Qualification and Signed Cross
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 6]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.
View answer and explanation
(Yes, 6, Yes)
Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, 6, Yes).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-coordinate-qualification · candidate
21. Candidate Labeled Vertex Qualification and Signed Cross
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 7]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.
View answer and explanation
(Yes, 7, Yes)
Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, 7, Yes).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-coordinate-qualification · candidate
22. Candidate Labeled Vertex Qualification and Signed Cross
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 8]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.
View answer and explanation
(Yes, 8, Yes)
Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, 8, Yes).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-coordinate-qualification · candidate
23. Candidate Labeled Vertex Qualification and Signed Cross
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 9]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.
View answer and explanation
(Yes, 9, Yes)
Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, 9, Yes).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-coordinate-qualification · candidate
24. Candidate Labeled Vertex Qualification and Signed Cross
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 10]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.
View answer and explanation
(Yes, 10, Yes)
Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, 10, Yes).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-coordinate-qualification · candidate
25. Candidate Labeled Vertex Qualification and Signed Cross
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 11]]. Give (are all three points distinct?, signed (B−A)×(C−A), are they distinct and noncollinear?), allowing degenerate candidates.
View answer and explanation
(Yes, 11, Yes)
Check distinctness and the exact signed determinant separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (Yes, 11, Yes).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-coordinate-qualification · candidate
26. Labeled Opposite Endpoints and Squared Side
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.
View answer and explanation
([2, 3], 1)
Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([2, 3], 1).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-opposite-side-squares · candidate
27. Labeled Opposite Endpoints and Squared Side
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [0, 1], [2, 0]]. The triangle is qualified as distinct and noncollinear, vertex=2. Give (other two vertex indices, opposite side’s squared L2 distance), not its distance.
View answer and explanation
([1, 3], 4)
Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([1, 3], 4).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-opposite-side-squares · candidate
28. Labeled Opposite Endpoints and Squared Side
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [3, 0], [0, 1]]. The triangle is qualified as distinct and noncollinear, vertex=3. Give (other two vertex indices, opposite side’s squared L2 distance), not its distance.
View answer and explanation
([1, 2], 9)
Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([1, 2], 9).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-opposite-side-squares · candidate
29. Labeled Opposite Endpoints and Squared Side
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 1], [0, 0], [4, 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.
View answer and explanation
([2, 3], 16)
Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([2, 3], 16).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-opposite-side-squares · candidate
30. Labeled Opposite Endpoints and Squared Side
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [0, 1], [5, 0]]. The triangle is qualified as distinct and noncollinear, vertex=2. Give (other two vertex indices, opposite side’s squared L2 distance), not its distance.
View answer and explanation
([1, 3], 25)
Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([1, 3], 25).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-opposite-side-squares · candidate
31. Labeled Opposite Endpoints and Squared Side
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [6, 0], [0, 1]]. The triangle is qualified as distinct and noncollinear, vertex=3. Give (other two vertex indices, opposite side’s squared L2 distance), not its distance.
View answer and explanation
([1, 2], 36)
Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([1, 2], 36).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-opposite-side-squares · candidate
32. Labeled Opposite Endpoints and Squared Side
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 1], [0, 0], [7, 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.
View answer and explanation
([2, 3], 49)
Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([2, 3], 49).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-opposite-side-squares · candidate
33. Labeled Opposite Endpoints and Squared Side
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [0, 1], [8, 0]]. The triangle is qualified as distinct and noncollinear, vertex=2. Give (other two vertex indices, opposite side’s squared L2 distance), not its distance.
View answer and explanation
([1, 3], 64)
Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([1, 3], 64).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-opposite-side-squares · candidate
34. Labeled Opposite Endpoints and Squared Side
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [9, 0], [0, 1]]. The triangle is qualified as distinct and noncollinear, vertex=3. Give (other two vertex indices, opposite side’s squared L2 distance), not its distance.
View answer and explanation
([1, 2], 81)
Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([1, 2], 81).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-opposite-side-squares · candidate
35. Labeled Opposite Endpoints and Squared Side
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 1], [0, 0], [10, 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.
View answer and explanation
([2, 3], 100)
Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([2, 3], 100).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-opposite-side-squares · candidate
36. Labeled Opposite Endpoints and Squared Side
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [0, 1], [11, 0]]. The triangle is qualified as distinct and noncollinear, vertex=2. Give (other two vertex indices, opposite side’s squared L2 distance), not its distance.
View answer and explanation
([1, 3], 121)
Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([1, 3], 121).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-opposite-side-squares · candidate
37. Labeled Opposite Endpoints and Squared Side
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [12, 0], [0, 1]]. The triangle is qualified as distinct and noncollinear, vertex=3. Give (other two vertex indices, opposite side’s squared L2 distance), not its distance.
View answer and explanation
([1, 2], 144)
Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([1, 2], 144).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-opposite-side-squares · candidate
38. Labeled Opposite Endpoints and Squared Side
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 1], [0, 0], [13, 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.
View answer and explanation
([2, 3], 169)
Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([2, 3], 169).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-opposite-side-squares · candidate
39. Labeled Opposite Endpoints and Squared Side
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [0, 1], [14, 0]]. The triangle is qualified as distinct and noncollinear, vertex=2. Give (other two vertex indices, opposite side’s squared L2 distance), not its distance.
View answer and explanation
([1, 3], 196)
Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([1, 3], 196).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-opposite-side-squares · candidate
40. Labeled Opposite Endpoints and Squared Side
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [15, 0], [0, 1]]. The triangle is qualified as distinct and noncollinear, vertex=3. Give (other two vertex indices, opposite side’s squared L2 distance), not its distance.
View answer and explanation
([1, 2], 225)
Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([1, 2], 225).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-opposite-side-squares · candidate
41. Labeled Opposite Endpoints and Squared Side
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 1], [0, 0], [16, 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.
View answer and explanation
([2, 3], 256)
Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([2, 3], 256).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-opposite-side-squares · candidate
42. Labeled Opposite Endpoints and Squared Side
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [0, 1], [17, 0]]. The triangle is qualified as distinct and noncollinear, vertex=2. Give (other two vertex indices, opposite side’s squared L2 distance), not its distance.
View answer and explanation
([1, 3], 289)
Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([1, 3], 289).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-opposite-side-squares · candidate
43. Labeled Opposite Endpoints and Squared Side
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [18, 0], [0, 1]]. The triangle is qualified as distinct and noncollinear, vertex=3. Give (other two vertex indices, opposite side’s squared L2 distance), not its distance.
View answer and explanation
([1, 2], 324)
Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([1, 2], 324).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-opposite-side-squares · candidate
44. Labeled Opposite Endpoints and Squared Side
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 1], [0, 0], [19, 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.
View answer and explanation
([2, 3], 361)
Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([2, 3], 361).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-opposite-side-squares · candidate
45. Labeled Opposite Endpoints and Squared Side
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [0, 1], [20, 0]]. The triangle is qualified as distinct and noncollinear, vertex=2. Give (other two vertex indices, opposite side’s squared L2 distance), not its distance.
View answer and explanation
([1, 3], 400)
Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([1, 3], 400).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-opposite-side-squares · candidate
46. Labeled Opposite Endpoints and Squared Side
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [21, 0], [0, 1]]. The triangle is qualified as distinct and noncollinear, vertex=3. Give (other two vertex indices, opposite side’s squared L2 distance), not its distance.
View answer and explanation
([1, 2], 441)
Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([1, 2], 441).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-opposite-side-squares · candidate
47. Labeled Opposite Endpoints and Squared Side
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 1], [0, 0], [22, 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.
View answer and explanation
([2, 3], 484)
Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([2, 3], 484).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-opposite-side-squares · candidate
48. Labeled Opposite Endpoints and Squared Side
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [0, 1], [23, 0]]. The triangle is qualified as distinct and noncollinear, vertex=2. Give (other two vertex indices, opposite side’s squared L2 distance), not its distance.
View answer and explanation
([1, 3], 529)
Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([1, 3], 529).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-opposite-side-squares · candidate
49. Labeled Opposite Endpoints and Squared Side
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [24, 0], [0, 1]]. The triangle is qualified as distinct and noncollinear, vertex=3. Give (other two vertex indices, opposite side’s squared L2 distance), not its distance.
View answer and explanation
([1, 2], 576)
Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([1, 2], 576).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-opposite-side-squares · candidate
50. Labeled Opposite Endpoints and Squared Side
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 1], [0, 0], [25, 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.
View answer and explanation
([2, 3], 625)
Select the other labels in order, then sum their coordinate-difference squares. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is ([2, 3], 625).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-opposite-side-squares · candidate
51. Cyclic versus Swapped Signed Coordinate Crosses
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -12]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.
View answer and explanation
(-12, -12, 12)
Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (-12, -12, 12).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-cyclic-label-orientation · candidate
52. Cyclic versus Swapped Signed Coordinate Crosses
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -11]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.
View answer and explanation
(-11, -11, 11)
Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (-11, -11, 11).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-cyclic-label-orientation · candidate
53. Cyclic versus Swapped Signed Coordinate Crosses
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -10]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.
View answer and explanation
(-10, -10, 10)
Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (-10, -10, 10).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-cyclic-label-orientation · candidate
54. Cyclic versus Swapped Signed Coordinate Crosses
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -9]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.
View answer and explanation
(-9, -9, 9)
Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (-9, -9, 9).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-cyclic-label-orientation · candidate
55. Cyclic versus Swapped Signed Coordinate Crosses
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -8]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.
View answer and explanation
(-8, -8, 8)
Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (-8, -8, 8).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-cyclic-label-orientation · candidate
56. Cyclic versus Swapped Signed Coordinate Crosses
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -7]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.
View answer and explanation
(-7, -7, 7)
Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (-7, -7, 7).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-cyclic-label-orientation · candidate
57. Cyclic versus Swapped Signed Coordinate Crosses
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -6]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.
View answer and explanation
(-6, -6, 6)
Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (-6, -6, 6).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-cyclic-label-orientation · candidate
58. Cyclic versus Swapped Signed Coordinate Crosses
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -5]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.
View answer and explanation
(-5, -5, 5)
Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (-5, -5, 5).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-cyclic-label-orientation · candidate
59. Cyclic versus Swapped Signed Coordinate Crosses
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -4]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.
View answer and explanation
(-4, -4, 4)
Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (-4, -4, 4).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-cyclic-label-orientation · candidate
60. Cyclic versus Swapped Signed Coordinate Crosses
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -3]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.
View answer and explanation
(-3, -3, 3)
Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (-3, -3, 3).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-cyclic-label-orientation · candidate
61. Cyclic versus Swapped Signed Coordinate Crosses
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -2]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.
View answer and explanation
(-2, -2, 2)
Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (-2, -2, 2).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-cyclic-label-orientation · candidate
62. Cyclic versus Swapped Signed Coordinate Crosses
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -1]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.
View answer and explanation
(-1, -1, 1)
Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (-1, -1, 1).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-cyclic-label-orientation · candidate
63. Cyclic versus Swapped Signed Coordinate Crosses
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 1]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.
View answer and explanation
(1, 1, -1)
Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1, 1, -1).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-cyclic-label-orientation · candidate
64. Cyclic versus Swapped Signed Coordinate Crosses
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 2]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.
View answer and explanation
(2, 2, -2)
Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (2, 2, -2).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-cyclic-label-orientation · candidate
65. Cyclic versus Swapped Signed Coordinate Crosses
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 3]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.
View answer and explanation
(3, 3, -3)
Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (3, 3, -3).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-cyclic-label-orientation · candidate
66. Cyclic versus Swapped Signed Coordinate Crosses
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 4]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.
View answer and explanation
(4, 4, -4)
Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (4, 4, -4).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-cyclic-label-orientation · candidate
67. Cyclic versus Swapped Signed Coordinate Crosses
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 5]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.
View answer and explanation
(5, 5, -5)
Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (5, 5, -5).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-cyclic-label-orientation · candidate
68. Cyclic versus Swapped Signed Coordinate Crosses
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 6]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.
View answer and explanation
(6, 6, -6)
Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (6, 6, -6).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-cyclic-label-orientation · candidate
69. Cyclic versus Swapped Signed Coordinate Crosses
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 7]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.
View answer and explanation
(7, 7, -7)
Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (7, 7, -7).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-cyclic-label-orientation · candidate
70. Cyclic versus Swapped Signed Coordinate Crosses
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 8]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.
View answer and explanation
(8, 8, -8)
Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (8, 8, -8).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-cyclic-label-orientation · candidate
71. Cyclic versus Swapped Signed Coordinate Crosses
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 9]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.
View answer and explanation
(9, 9, -9)
Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (9, 9, -9).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-cyclic-label-orientation · candidate
72. Cyclic versus Swapped Signed Coordinate Crosses
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 10]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.
View answer and explanation
(10, 10, -10)
Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (10, 10, -10).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-cyclic-label-orientation · candidate
73. Cyclic versus Swapped Signed Coordinate Crosses
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 11]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.
View answer and explanation
(11, 11, -11)
Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (11, 11, -11).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-cyclic-label-orientation · candidate
74. Cyclic versus Swapped Signed Coordinate Crosses
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 12]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.
View answer and explanation
(12, 12, -12)
Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (12, 12, -12).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-cyclic-label-orientation · candidate
75. Cyclic versus Swapped Signed Coordinate Crosses
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, 13]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.
View answer and explanation
(13, 13, -13)
Compute the signed coordinate determinant in each label order; cycling and exchanging labels have different orientation effects. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (13, 13, -13).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-cyclic-label-orientation · candidate
76. Opposite Midpoint and Three Squared Segments
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 points are distinct and noncollinear, vertex=1, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 0/1, 1/4, 1/4, 5/4)
Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 1/4, 1/4, 5/4).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-median-equality · candidate
77. Opposite Midpoint and Three Squared Segments
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[-1, 0], [0, 2], [2, 0]]. The points are distinct and noncollinear, vertex=2, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 0/1, 9/4, 9/4, 17/4)
Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 9/4, 9/4, 17/4).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-median-equality · candidate
78. Opposite Midpoint and Three Squared Segments
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[-2, 0], [3, 0], [0, 3]]. The points are distinct and noncollinear, vertex=3, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 0/1, 25/4, 25/4, 37/4)
Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 25/4, 25/4, 37/4).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-median-equality · candidate
79. Opposite Midpoint and Three Squared Segments
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 4], [-3, 0], [4, 0]]. The points are distinct and noncollinear, vertex=1, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 0/1, 49/4, 49/4, 65/4)
Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 49/4, 49/4, 65/4).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-median-equality · candidate
80. Opposite Midpoint and Three Squared Segments
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[-4, 0], [0, 5], [5, 0]]. The points are distinct and noncollinear, vertex=2, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 0/1, 81/4, 81/4, 101/4)
Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 81/4, 81/4, 101/4).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-median-equality · candidate
81. Opposite Midpoint and Three Squared Segments
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[-5, 0], [6, 0], [0, 6]]. The points are distinct and noncollinear, vertex=3, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 0/1, 121/4, 121/4, 145/4)
Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 121/4, 121/4, 145/4).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-median-equality · candidate
82. Opposite Midpoint and Three Squared Segments
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 7], [-6, 0], [7, 0]]. The points are distinct and noncollinear, vertex=1, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 0/1, 169/4, 169/4, 197/4)
Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 169/4, 169/4, 197/4).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-median-equality · candidate
83. Opposite Midpoint and Three Squared Segments
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[-7, 0], [0, 8], [8, 0]]. The points are distinct and noncollinear, vertex=2, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 0/1, 225/4, 225/4, 257/4)
Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 225/4, 225/4, 257/4).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-median-equality · candidate
84. Opposite Midpoint and Three Squared Segments
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[-8, 0], [9, 0], [0, 9]]. The points are distinct and noncollinear, vertex=3, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 0/1, 289/4, 289/4, 325/4)
Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 289/4, 289/4, 325/4).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-median-equality · candidate
85. Opposite Midpoint and Three Squared Segments
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 10], [-9, 0], [10, 0]]. The points are distinct and noncollinear, vertex=1, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 0/1, 361/4, 361/4, 401/4)
Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 361/4, 361/4, 401/4).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-median-equality · candidate
86. Opposite Midpoint and Three Squared Segments
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[-10, 0], [0, 11], [11, 0]]. The points are distinct and noncollinear, vertex=2, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 0/1, 441/4, 441/4, 485/4)
Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 441/4, 441/4, 485/4).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-median-equality · candidate
87. Opposite Midpoint and Three Squared Segments
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[-11, 0], [12, 0], [0, 12]]. The points are distinct and noncollinear, vertex=3, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 0/1, 529/4, 529/4, 577/4)
Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 529/4, 529/4, 577/4).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-median-equality · candidate
88. Opposite Midpoint and Three Squared Segments
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 13], [-12, 0], [13, 0]]. The points are distinct and noncollinear, vertex=1, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 0/1, 625/4, 625/4, 677/4)
Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 625/4, 625/4, 677/4).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-median-equality · candidate
89. Opposite Midpoint and Three Squared Segments
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[-13, 0], [0, 14], [14, 0]]. The points are distinct and noncollinear, vertex=2, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 0/1, 729/4, 729/4, 785/4)
Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 729/4, 729/4, 785/4).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-median-equality · candidate
90. Opposite Midpoint and Three Squared Segments
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[-14, 0], [15, 0], [0, 15]]. The points are distinct and noncollinear, vertex=3, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 0/1, 841/4, 841/4, 901/4)
Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 841/4, 841/4, 901/4).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-median-equality · candidate
91. Opposite Midpoint and Three Squared Segments
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 16], [-15, 0], [16, 0]]. The points are distinct and noncollinear, vertex=1, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 0/1, 961/4, 961/4, 1025/4)
Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 961/4, 961/4, 1025/4).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-median-equality · candidate
92. Opposite Midpoint and Three Squared Segments
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[-16, 0], [0, 17], [17, 0]]. The points are distinct and noncollinear, vertex=2, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 0/1, 1089/4, 1089/4, 1157/4)
Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 1089/4, 1089/4, 1157/4).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-median-equality · candidate
93. Opposite Midpoint and Three Squared Segments
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[-17, 0], [18, 0], [0, 18]]. The points are distinct and noncollinear, vertex=3, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 0/1, 1225/4, 1225/4, 1297/4)
Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 1225/4, 1225/4, 1297/4).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-median-equality · candidate
94. Opposite Midpoint and Three Squared Segments
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 19], [-18, 0], [19, 0]]. The points are distinct and noncollinear, vertex=1, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 0/1, 1369/4, 1369/4, 1445/4)
Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 1369/4, 1369/4, 1445/4).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-median-equality · candidate
95. Opposite Midpoint and Three Squared Segments
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[-19, 0], [0, 20], [20, 0]]. The points are distinct and noncollinear, vertex=2, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 0/1, 1521/4, 1521/4, 1601/4)
Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 1521/4, 1521/4, 1601/4).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-median-equality · candidate
96. Opposite Midpoint and Three Squared Segments
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[-20, 0], [21, 0], [0, 21]]. The points are distinct and noncollinear, vertex=3, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 0/1, 1681/4, 1681/4, 1765/4)
Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 1681/4, 1681/4, 1765/4).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-median-equality · candidate
97. Opposite Midpoint and Three Squared Segments
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 22], [-21, 0], [22, 0]]. The points are distinct and noncollinear, vertex=1, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 0/1, 1849/4, 1849/4, 1937/4)
Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 1849/4, 1849/4, 1937/4).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-median-equality · candidate
98. Opposite Midpoint and Three Squared Segments
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[-22, 0], [0, 23], [23, 0]]. The points are distinct and noncollinear, vertex=2, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 0/1, 2025/4, 2025/4, 2117/4)
Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 2025/4, 2025/4, 2117/4).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-median-equality · candidate
99. Opposite Midpoint and Three Squared Segments
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[-23, 0], [24, 0], [0, 24]]. The points are distinct and noncollinear, vertex=3, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 0/1, 2209/4, 2209/4, 2305/4)
Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 2209/4, 2209/4, 2305/4).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-median-equality · candidate
100. Opposite Midpoint and Three Squared Segments
In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 25], [-24, 0], [25, 0]]. The points are distinct and noncollinear, vertex=1, and M is the opposite side’s midpoint. Give (Mx, My, first opposite endpoint-to-M square, second endpoint-to-M square, vertex-to-M square). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 0/1, 2401/4, 2401/4, 2501/4)
Average opposite endpoint coordinates, then compute each exact squared segment separately. Finite coordinate checks are not a general geometry proof; do not read measurements from a sketch. The exact result, retaining the requested order, is (1/2, 0/1, 2401/4, 2401/4, 2501/4).
Reviewed step-by-step derivation
Triangle · qf-density13-triangle-median-equality · candidate
Canonical question JSONL · English JSONL · Validation report