# 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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0001/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0002/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0003/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0004/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0005/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0006/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0007/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0008/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0009/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0010/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0011/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0012/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0013/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0014/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0015/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0016/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0017/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0018/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0019/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0020/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0021/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0022/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0023/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0024/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0025/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0026/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0027/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0028/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0029/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0030/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0031/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0032/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0033/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0034/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0035/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0036/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0037/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0038/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0039/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0040/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0041/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0042/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0043/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0044/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0045/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0046/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0047/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0048/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0049/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0050/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0051/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0052/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0053/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0054/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0055/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0056/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0057/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0058/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0059/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0060/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0061/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0062/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0063/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0064/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0065/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0066/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0067/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0068/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0069/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0070/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0071/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0072/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0073/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0074/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0075/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0076/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0077/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0078/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0079/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0080/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0081/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0082/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0083/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0084/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0085/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0086/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0087/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0088/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0089/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0090/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0091/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0092/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0093/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0094/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0095/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0096/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0097/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0098/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0099/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r07-triangle-0100/)

Triangle · qf-density13-triangle-median-equality · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261009-advance-20261013-r07-triangle-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261009-advance-20261013-r07-triangle-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261009-advance-20261013-r07-triangle-foundations/validation-report.json)