# Triangle Object Language

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-20260925-triangle-object-language · 2026-09-25 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Triangle Validity from Explicit Conditions

The three points in order are A, A, C; the prompt explicitly states that they are collinear. Using only the distinctness and noncollinearity conditions, do they form a nondegenerate triangle?

View answer and explanationNo

The vertices are not all distinct, so they do not form the nondegenerate triangle of this entry.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-explicit-validity-0001/)

Triangle · qf-triangle-explicit-validity · candidate





## 2. Triangle Validity from Explicit Conditions

The three points in order are D, E, F; the prompt explicitly states that they are noncollinear. Using only the distinctness and noncollinearity conditions, do they form a nondegenerate triangle?

View answer and explanationYes

The three vertices are distinct and noncollinear, so they form a nondegenerate triangle.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-explicit-validity-0002/)

Triangle · qf-triangle-explicit-validity · candidate





## 3. Triangle Validity from Explicit Conditions

The three points in order are G, H, I; the prompt explicitly states that they are noncollinear. Using only the distinctness and noncollinearity conditions, do they form a nondegenerate triangle?

View answer and explanationYes

The three vertices are distinct and noncollinear, so they form a nondegenerate triangle.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-explicit-validity-0003/)

Triangle · qf-triangle-explicit-validity · candidate





## 4. Triangle Validity from Explicit Conditions

The three points in order are J, K, L; the prompt explicitly states that they are noncollinear. Using only the distinctness and noncollinearity conditions, do they form a nondegenerate triangle?

View answer and explanationYes

The three vertices are distinct and noncollinear, so they form a nondegenerate triangle.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-explicit-validity-0004/)

Triangle · qf-triangle-explicit-validity · candidate





## 5. Triangle Validity from Explicit Conditions

The three points in order are M, M, P; the prompt explicitly states that they are collinear. Using only the distinctness and noncollinearity conditions, do they form a nondegenerate triangle?

View answer and explanationNo

The vertices are not all distinct, so they do not form the nondegenerate triangle of this entry.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-explicit-validity-0005/)

Triangle · qf-triangle-explicit-validity · candidate





## 6. Triangle Validity from Explicit Conditions

The three points in order are Q, R, S; the prompt explicitly states that they are collinear. Using only the distinctness and noncollinearity conditions, do they form a nondegenerate triangle?

View answer and explanationNo

The vertices are distinct but collinear, so they do not form the nondegenerate triangle of this entry.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-explicit-validity-0006/)

Triangle · qf-triangle-explicit-validity · candidate





## 7. Triangle Validity from Explicit Conditions

The three points in order are T, U, V; the prompt explicitly states that they are noncollinear. Using only the distinctness and noncollinearity conditions, do they form a nondegenerate triangle?

View answer and explanationYes

The three vertices are distinct and noncollinear, so they form a nondegenerate triangle.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-explicit-validity-0007/)

Triangle · qf-triangle-explicit-validity · candidate





## 8. Triangle Validity from Explicit Conditions

The three points in order are W, X, Y; the prompt explicitly states that they are noncollinear. Using only the distinctness and noncollinearity conditions, do they form a nondegenerate triangle?

View answer and explanationYes

The three vertices are distinct and noncollinear, so they form a nondegenerate triangle.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-explicit-validity-0008/)

Triangle · qf-triangle-explicit-validity · candidate





## 9. Triangle Validity from Explicit Conditions

The three points in order are A, A, D; the prompt explicitly states that they are collinear. Using only the distinctness and noncollinearity conditions, do they form a nondegenerate triangle?

View answer and explanationNo

The vertices are not all distinct, so they do not form the nondegenerate triangle of this entry.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-explicit-validity-0009/)

Triangle · qf-triangle-explicit-validity · candidate





## 10. Triangle Validity from Explicit Conditions

The three points in order are A, C, E; the prompt explicitly states that they are noncollinear. Using only the distinctness and noncollinearity conditions, do they form a nondegenerate triangle?

View answer and explanationYes

The three vertices are distinct and noncollinear, so they form a nondegenerate triangle.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-explicit-validity-0010/)

Triangle · qf-triangle-explicit-validity · candidate





## 11. Triangle Validity from Explicit Conditions

The three points in order are B, D, F; the prompt explicitly states that they are collinear. Using only the distinctness and noncollinearity conditions, do they form a nondegenerate triangle?

View answer and explanationNo

The vertices are distinct but collinear, so they do not form the nondegenerate triangle of this entry.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-explicit-validity-0011/)

Triangle · qf-triangle-explicit-validity · candidate





## 12. Triangle Validity from Explicit Conditions

The three points in order are C, E, G; the prompt explicitly states that they are noncollinear. Using only the distinctness and noncollinearity conditions, do they form a nondegenerate triangle?

View answer and explanationYes

The three vertices are distinct and noncollinear, so they form a nondegenerate triangle.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-explicit-validity-0012/)

Triangle · qf-triangle-explicit-validity · candidate





## 13. Triangle Validity from Explicit Conditions

The three points in order are D, D, H; the prompt explicitly states that they are collinear. Using only the distinctness and noncollinearity conditions, do they form a nondegenerate triangle?

View answer and explanationNo

The vertices are not all distinct, so they do not form the nondegenerate triangle of this entry.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-explicit-validity-0013/)

Triangle · qf-triangle-explicit-validity · candidate





## 14. Triangle Validity from Explicit Conditions

The three points in order are E, G, I; the prompt explicitly states that they are noncollinear. Using only the distinctness and noncollinearity conditions, do they form a nondegenerate triangle?

View answer and explanationYes

The three vertices are distinct and noncollinear, so they form a nondegenerate triangle.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-explicit-validity-0014/)

Triangle · qf-triangle-explicit-validity · candidate





## 15. Triangle Validity from Explicit Conditions

The three points in order are F, H, J; the prompt explicitly states that they are noncollinear. Using only the distinctness and noncollinearity conditions, do they form a nondegenerate triangle?

View answer and explanationYes

The three vertices are distinct and noncollinear, so they form a nondegenerate triangle.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-explicit-validity-0015/)

Triangle · qf-triangle-explicit-validity · candidate





## 16. Triangle Validity from Explicit Conditions

The three points in order are G, I, K; the prompt explicitly states that they are collinear. Using only the distinctness and noncollinearity conditions, do they form a nondegenerate triangle?

View answer and explanationNo

The vertices are distinct but collinear, so they do not form the nondegenerate triangle of this entry.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-explicit-validity-0016/)

Triangle · qf-triangle-explicit-validity · candidate





## 17. Triangle Validity from Explicit Conditions

The three points in order are H, H, L; the prompt explicitly states that they are collinear. Using only the distinctness and noncollinearity conditions, do they form a nondegenerate triangle?

View answer and explanationNo

The vertices are not all distinct, so they do not form the nondegenerate triangle of this entry.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-explicit-validity-0017/)

Triangle · qf-triangle-explicit-validity · candidate





## 18. Triangle Validity from Explicit Conditions

The three points in order are I, K, M; the prompt explicitly states that they are noncollinear. Using only the distinctness and noncollinearity conditions, do they form a nondegenerate triangle?

View answer and explanationYes

The three vertices are distinct and noncollinear, so they form a nondegenerate triangle.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-explicit-validity-0018/)

Triangle · qf-triangle-explicit-validity · candidate





## 19. Triangle Validity from Explicit Conditions

The three points in order are J, L, N; the prompt explicitly states that they are noncollinear. Using only the distinctness and noncollinearity conditions, do they form a nondegenerate triangle?

View answer and explanationYes

The three vertices are distinct and noncollinear, so they form a nondegenerate triangle.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-explicit-validity-0019/)

Triangle · qf-triangle-explicit-validity · candidate





## 20. Triangle Validity from Explicit Conditions

The three points in order are K, M, O; the prompt explicitly states that they are noncollinear. Using only the distinctness and noncollinearity conditions, do they form a nondegenerate triangle?

View answer and explanationYes

The three vertices are distinct and noncollinear, so they form a nondegenerate triangle.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-explicit-validity-0020/)

Triangle · qf-triangle-explicit-validity · candidate





## 21. Triangle Validity from Explicit Conditions

The three points in order are L, L, P; the prompt explicitly states that they are collinear. Using only the distinctness and noncollinearity conditions, do they form a nondegenerate triangle?

View answer and explanationNo

The vertices are not all distinct, so they do not form the nondegenerate triangle of this entry.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-explicit-validity-0021/)

Triangle · qf-triangle-explicit-validity · candidate





## 22. Triangle Validity from Explicit Conditions

The three points in order are M, O, Q; the prompt explicitly states that they are noncollinear. Using only the distinctness and noncollinearity conditions, do they form a nondegenerate triangle?

View answer and explanationYes

The three vertices are distinct and noncollinear, so they form a nondegenerate triangle.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-explicit-validity-0022/)

Triangle · qf-triangle-explicit-validity · candidate





## 23. Triangle Validity from Explicit Conditions

The three points in order are N, P, R; the prompt explicitly states that they are noncollinear. Using only the distinctness and noncollinearity conditions, do they form a nondegenerate triangle?

View answer and explanationYes

The three vertices are distinct and noncollinear, so they form a nondegenerate triangle.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-explicit-validity-0023/)

Triangle · qf-triangle-explicit-validity · candidate





## 24. Triangle Validity from Explicit Conditions

The three points in order are O, Q, S; the prompt explicitly states that they are noncollinear. Using only the distinctness and noncollinearity conditions, do they form a nondegenerate triangle?

View answer and explanationYes

The three vertices are distinct and noncollinear, so they form a nondegenerate triangle.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-explicit-validity-0024/)

Triangle · qf-triangle-explicit-validity · candidate





## 25. Triangle Validity from Explicit Conditions

The three points in order are P, P, T; the prompt explicitly states that they are collinear. Using only the distinctness and noncollinearity conditions, do they form a nondegenerate triangle?

View answer and explanationNo

The vertices are not all distinct, so they do not form the nondegenerate triangle of this entry.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-explicit-validity-0025/)

Triangle · qf-triangle-explicit-validity · candidate





## 26. Vertex Set of a Triangle

Triangle ABC has explicitly distinct noncollinear vertices. Write its vertex set in canonical order.

View answer and explanation{A, B, C}

Taking the three labels and sorting them canonically gives {A, B, C}.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-vertex-set-0001/)

Triangle · qf-triangle-vertex-set · candidate





## 27. Vertex Set of a Triangle

Triangle DEF has explicitly distinct noncollinear vertices. Write its vertex set in canonical order.

View answer and explanation{D, E, F}

Taking the three labels and sorting them canonically gives {D, E, F}.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-vertex-set-0002/)

Triangle · qf-triangle-vertex-set · candidate





## 28. Vertex Set of a Triangle

Triangle GHI has explicitly distinct noncollinear vertices. Write its vertex set in canonical order.

View answer and explanation{G, H, I}

Taking the three labels and sorting them canonically gives {G, H, I}.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-vertex-set-0003/)

Triangle · qf-triangle-vertex-set · candidate





## 29. Vertex Set of a Triangle

Triangle JKL has explicitly distinct noncollinear vertices. Write its vertex set in canonical order.

View answer and explanation{J, K, L}

Taking the three labels and sorting them canonically gives {J, K, L}.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-vertex-set-0004/)

Triangle · qf-triangle-vertex-set · candidate





## 30. Vertex Set of a Triangle

Triangle MNP has explicitly distinct noncollinear vertices. Write its vertex set in canonical order.

View answer and explanation{M, N, P}

Taking the three labels and sorting them canonically gives {M, N, P}.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-vertex-set-0005/)

Triangle · qf-triangle-vertex-set · candidate





## 31. Vertex Set of a Triangle

Triangle QRS has explicitly distinct noncollinear vertices. Write its vertex set in canonical order.

View answer and explanation{Q, R, S}

Taking the three labels and sorting them canonically gives {Q, R, S}.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-vertex-set-0006/)

Triangle · qf-triangle-vertex-set · candidate





## 32. Vertex Set of a Triangle

Triangle TUV has explicitly distinct noncollinear vertices. Write its vertex set in canonical order.

View answer and explanation{T, U, V}

Taking the three labels and sorting them canonically gives {T, U, V}.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-vertex-set-0007/)

Triangle · qf-triangle-vertex-set · candidate





## 33. Vertex Set of a Triangle

Triangle WXY has explicitly distinct noncollinear vertices. Write its vertex set in canonical order.

View answer and explanation{W, X, Y}

Taking the three labels and sorting them canonically gives {W, X, Y}.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-vertex-set-0008/)

Triangle · qf-triangle-vertex-set · candidate





## 34. Vertex Set of a Triangle

Triangle ABD has explicitly distinct noncollinear vertices. Write its vertex set in canonical order.

View answer and explanation{A, B, D}

Taking the three labels and sorting them canonically gives {A, B, D}.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-vertex-set-0009/)

Triangle · qf-triangle-vertex-set · candidate





## 35. Vertex Set of a Triangle

Triangle ACE has explicitly distinct noncollinear vertices. Write its vertex set in canonical order.

View answer and explanation{A, C, E}

Taking the three labels and sorting them canonically gives {A, C, E}.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-vertex-set-0010/)

Triangle · qf-triangle-vertex-set · candidate





## 36. Vertex Set of a Triangle

Triangle BDF has explicitly distinct noncollinear vertices. Write its vertex set in canonical order.

View answer and explanation{B, D, F}

Taking the three labels and sorting them canonically gives {B, D, F}.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-vertex-set-0011/)

Triangle · qf-triangle-vertex-set · candidate





## 37. Vertex Set of a Triangle

Triangle CEG has explicitly distinct noncollinear vertices. Write its vertex set in canonical order.

View answer and explanation{C, E, G}

Taking the three labels and sorting them canonically gives {C, E, G}.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-vertex-set-0012/)

Triangle · qf-triangle-vertex-set · candidate





## 38. Vertex Set of a Triangle

Triangle DFH has explicitly distinct noncollinear vertices. Write its vertex set in canonical order.

View answer and explanation{D, F, H}

Taking the three labels and sorting them canonically gives {D, F, H}.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-vertex-set-0013/)

Triangle · qf-triangle-vertex-set · candidate





## 39. Vertex Set of a Triangle

Triangle EGI has explicitly distinct noncollinear vertices. Write its vertex set in canonical order.

View answer and explanation{E, G, I}

Taking the three labels and sorting them canonically gives {E, G, I}.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-vertex-set-0014/)

Triangle · qf-triangle-vertex-set · candidate





## 40. Vertex Set of a Triangle

Triangle FHJ has explicitly distinct noncollinear vertices. Write its vertex set in canonical order.

View answer and explanation{F, H, J}

Taking the three labels and sorting them canonically gives {F, H, J}.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-vertex-set-0015/)

Triangle · qf-triangle-vertex-set · candidate





## 41. Vertex Set of a Triangle

Triangle GIK has explicitly distinct noncollinear vertices. Write its vertex set in canonical order.

View answer and explanation{G, I, K}

Taking the three labels and sorting them canonically gives {G, I, K}.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-vertex-set-0016/)

Triangle · qf-triangle-vertex-set · candidate





## 42. Vertex Set of a Triangle

Triangle HJL has explicitly distinct noncollinear vertices. Write its vertex set in canonical order.

View answer and explanation{H, J, L}

Taking the three labels and sorting them canonically gives {H, J, L}.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-vertex-set-0017/)

Triangle · qf-triangle-vertex-set · candidate





## 43. Vertex Set of a Triangle

Triangle IKM has explicitly distinct noncollinear vertices. Write its vertex set in canonical order.

View answer and explanation{I, K, M}

Taking the three labels and sorting them canonically gives {I, K, M}.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-vertex-set-0018/)

Triangle · qf-triangle-vertex-set · candidate





## 44. Vertex Set of a Triangle

Triangle JLN has explicitly distinct noncollinear vertices. Write its vertex set in canonical order.

View answer and explanation{J, L, N}

Taking the three labels and sorting them canonically gives {J, L, N}.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-vertex-set-0019/)

Triangle · qf-triangle-vertex-set · candidate





## 45. Vertex Set of a Triangle

Triangle KMO has explicitly distinct noncollinear vertices. Write its vertex set in canonical order.

View answer and explanation{K, M, O}

Taking the three labels and sorting them canonically gives {K, M, O}.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-vertex-set-0020/)

Triangle · qf-triangle-vertex-set · candidate





## 46. Vertex Set of a Triangle

Triangle LNP has explicitly distinct noncollinear vertices. Write its vertex set in canonical order.

View answer and explanation{L, N, P}

Taking the three labels and sorting them canonically gives {L, N, P}.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-vertex-set-0021/)

Triangle · qf-triangle-vertex-set · candidate





## 47. Vertex Set of a Triangle

Triangle MOQ has explicitly distinct noncollinear vertices. Write its vertex set in canonical order.

View answer and explanation{M, O, Q}

Taking the three labels and sorting them canonically gives {M, O, Q}.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-vertex-set-0022/)

Triangle · qf-triangle-vertex-set · candidate





## 48. Vertex Set of a Triangle

Triangle NPR has explicitly distinct noncollinear vertices. Write its vertex set in canonical order.

View answer and explanation{N, P, R}

Taking the three labels and sorting them canonically gives {N, P, R}.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-vertex-set-0023/)

Triangle · qf-triangle-vertex-set · candidate





## 49. Vertex Set of a Triangle

Triangle OQS has explicitly distinct noncollinear vertices. Write its vertex set in canonical order.

View answer and explanation{O, Q, S}

Taking the three labels and sorting them canonically gives {O, Q, S}.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-vertex-set-0024/)

Triangle · qf-triangle-vertex-set · candidate





## 50. Vertex Set of a Triangle

Triangle PRT has explicitly distinct noncollinear vertices. Write its vertex set in canonical order.

View answer and explanation{P, R, T}

Taking the three labels and sorting them canonically gives {P, R, T}.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-vertex-set-0025/)

Triangle · qf-triangle-vertex-set · candidate





## 51. Undirected Side Set of a Triangle

Triangle ABC has explicitly distinct noncollinear vertices. List its three undirected sides in canonical order.

View answer and explanationAB, AC, BC

Pairwise connection of the vertices followed by canonical ordering gives AB, AC, BC.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-side-set-0001/)

Triangle · qf-triangle-side-set · candidate





## 52. Undirected Side Set of a Triangle

Triangle DEF has explicitly distinct noncollinear vertices. List its three undirected sides in canonical order.

View answer and explanationDE, DF, EF

Pairwise connection of the vertices followed by canonical ordering gives DE, DF, EF.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-side-set-0002/)

Triangle · qf-triangle-side-set · candidate





## 53. Undirected Side Set of a Triangle

Triangle GHI has explicitly distinct noncollinear vertices. List its three undirected sides in canonical order.

View answer and explanationGH, GI, HI

Pairwise connection of the vertices followed by canonical ordering gives GH, GI, HI.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-side-set-0003/)

Triangle · qf-triangle-side-set · candidate





## 54. Undirected Side Set of a Triangle

Triangle JKL has explicitly distinct noncollinear vertices. List its three undirected sides in canonical order.

View answer and explanationJK, JL, KL

Pairwise connection of the vertices followed by canonical ordering gives JK, JL, KL.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-side-set-0004/)

Triangle · qf-triangle-side-set · candidate





## 55. Undirected Side Set of a Triangle

Triangle MNP has explicitly distinct noncollinear vertices. List its three undirected sides in canonical order.

View answer and explanationMN, MP, NP

Pairwise connection of the vertices followed by canonical ordering gives MN, MP, NP.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-side-set-0005/)

Triangle · qf-triangle-side-set · candidate





## 56. Undirected Side Set of a Triangle

Triangle QRS has explicitly distinct noncollinear vertices. List its three undirected sides in canonical order.

View answer and explanationQR, QS, RS

Pairwise connection of the vertices followed by canonical ordering gives QR, QS, RS.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-side-set-0006/)

Triangle · qf-triangle-side-set · candidate





## 57. Undirected Side Set of a Triangle

Triangle TUV has explicitly distinct noncollinear vertices. List its three undirected sides in canonical order.

View answer and explanationTU, TV, UV

Pairwise connection of the vertices followed by canonical ordering gives TU, TV, UV.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-side-set-0007/)

Triangle · qf-triangle-side-set · candidate





## 58. Undirected Side Set of a Triangle

Triangle WXY has explicitly distinct noncollinear vertices. List its three undirected sides in canonical order.

View answer and explanationWX, WY, XY

Pairwise connection of the vertices followed by canonical ordering gives WX, WY, XY.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-side-set-0008/)

Triangle · qf-triangle-side-set · candidate





## 59. Undirected Side Set of a Triangle

Triangle ABD has explicitly distinct noncollinear vertices. List its three undirected sides in canonical order.

View answer and explanationAB, AD, BD

Pairwise connection of the vertices followed by canonical ordering gives AB, AD, BD.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-side-set-0009/)

Triangle · qf-triangle-side-set · candidate





## 60. Undirected Side Set of a Triangle

Triangle ACE has explicitly distinct noncollinear vertices. List its three undirected sides in canonical order.

View answer and explanationAC, AE, CE

Pairwise connection of the vertices followed by canonical ordering gives AC, AE, CE.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-side-set-0010/)

Triangle · qf-triangle-side-set · candidate





## 61. Undirected Side Set of a Triangle

Triangle BDF has explicitly distinct noncollinear vertices. List its three undirected sides in canonical order.

View answer and explanationBD, BF, DF

Pairwise connection of the vertices followed by canonical ordering gives BD, BF, DF.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-side-set-0011/)

Triangle · qf-triangle-side-set · candidate





## 62. Undirected Side Set of a Triangle

Triangle CEG has explicitly distinct noncollinear vertices. List its three undirected sides in canonical order.

View answer and explanationCE, CG, EG

Pairwise connection of the vertices followed by canonical ordering gives CE, CG, EG.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-side-set-0012/)

Triangle · qf-triangle-side-set · candidate





## 63. Undirected Side Set of a Triangle

Triangle DFH has explicitly distinct noncollinear vertices. List its three undirected sides in canonical order.

View answer and explanationDF, DH, FH

Pairwise connection of the vertices followed by canonical ordering gives DF, DH, FH.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-side-set-0013/)

Triangle · qf-triangle-side-set · candidate





## 64. Undirected Side Set of a Triangle

Triangle EGI has explicitly distinct noncollinear vertices. List its three undirected sides in canonical order.

View answer and explanationEG, EI, GI

Pairwise connection of the vertices followed by canonical ordering gives EG, EI, GI.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-side-set-0014/)

Triangle · qf-triangle-side-set · candidate





## 65. Undirected Side Set of a Triangle

Triangle FHJ has explicitly distinct noncollinear vertices. List its three undirected sides in canonical order.

View answer and explanationFH, FJ, HJ

Pairwise connection of the vertices followed by canonical ordering gives FH, FJ, HJ.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-side-set-0015/)

Triangle · qf-triangle-side-set · candidate





## 66. Undirected Side Set of a Triangle

Triangle GIK has explicitly distinct noncollinear vertices. List its three undirected sides in canonical order.

View answer and explanationGI, GK, IK

Pairwise connection of the vertices followed by canonical ordering gives GI, GK, IK.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-side-set-0016/)

Triangle · qf-triangle-side-set · candidate





## 67. Undirected Side Set of a Triangle

Triangle HJL has explicitly distinct noncollinear vertices. List its three undirected sides in canonical order.

View answer and explanationHJ, HL, JL

Pairwise connection of the vertices followed by canonical ordering gives HJ, HL, JL.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-side-set-0017/)

Triangle · qf-triangle-side-set · candidate





## 68. Undirected Side Set of a Triangle

Triangle IKM has explicitly distinct noncollinear vertices. List its three undirected sides in canonical order.

View answer and explanationIK, IM, KM

Pairwise connection of the vertices followed by canonical ordering gives IK, IM, KM.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-side-set-0018/)

Triangle · qf-triangle-side-set · candidate





## 69. Undirected Side Set of a Triangle

Triangle JLN has explicitly distinct noncollinear vertices. List its three undirected sides in canonical order.

View answer and explanationJL, JN, LN

Pairwise connection of the vertices followed by canonical ordering gives JL, JN, LN.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-side-set-0019/)

Triangle · qf-triangle-side-set · candidate





## 70. Undirected Side Set of a Triangle

Triangle KMO has explicitly distinct noncollinear vertices. List its three undirected sides in canonical order.

View answer and explanationKM, KO, MO

Pairwise connection of the vertices followed by canonical ordering gives KM, KO, MO.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-side-set-0020/)

Triangle · qf-triangle-side-set · candidate





## 71. Undirected Side Set of a Triangle

Triangle LNP has explicitly distinct noncollinear vertices. List its three undirected sides in canonical order.

View answer and explanationLN, LP, NP

Pairwise connection of the vertices followed by canonical ordering gives LN, LP, NP.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-side-set-0021/)

Triangle · qf-triangle-side-set · candidate





## 72. Undirected Side Set of a Triangle

Triangle MOQ has explicitly distinct noncollinear vertices. List its three undirected sides in canonical order.

View answer and explanationMO, MQ, OQ

Pairwise connection of the vertices followed by canonical ordering gives MO, MQ, OQ.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-side-set-0022/)

Triangle · qf-triangle-side-set · candidate





## 73. Undirected Side Set of a Triangle

Triangle NPR has explicitly distinct noncollinear vertices. List its three undirected sides in canonical order.

View answer and explanationNP, NR, PR

Pairwise connection of the vertices followed by canonical ordering gives NP, NR, PR.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-side-set-0023/)

Triangle · qf-triangle-side-set · candidate





## 74. Undirected Side Set of a Triangle

Triangle OQS has explicitly distinct noncollinear vertices. List its three undirected sides in canonical order.

View answer and explanationOQ, OS, QS

Pairwise connection of the vertices followed by canonical ordering gives OQ, OS, QS.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-side-set-0024/)

Triangle · qf-triangle-side-set · candidate





## 75. Undirected Side Set of a Triangle

Triangle PRT has explicitly distinct noncollinear vertices. List its three undirected sides in canonical order.

View answer and explanationPR, PT, RT

Pairwise connection of the vertices followed by canonical ordering gives PR, PT, RT.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-side-set-0025/)

Triangle · qf-triangle-side-set · candidate





## 76. Side Opposite a Vertex

Triangle ABC has explicitly distinct noncollinear vertices. Which undirected side is opposite vertex A?

View answer and explanationBC

Removing vertex A leaves the other two vertices, which form side BC.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-opposite-side-0001/)

Triangle · qf-triangle-opposite-side · candidate





## 77. Side Opposite a Vertex

Triangle DEF has explicitly distinct noncollinear vertices. Which undirected side is opposite vertex E?

View answer and explanationDF

Removing vertex E leaves the other two vertices, which form side DF.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-opposite-side-0002/)

Triangle · qf-triangle-opposite-side · candidate





## 78. Side Opposite a Vertex

Triangle GHI has explicitly distinct noncollinear vertices. Which undirected side is opposite vertex I?

View answer and explanationGH

Removing vertex I leaves the other two vertices, which form side GH.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-opposite-side-0003/)

Triangle · qf-triangle-opposite-side · candidate





## 79. Side Opposite a Vertex

Triangle JKL has explicitly distinct noncollinear vertices. Which undirected side is opposite vertex J?

View answer and explanationKL

Removing vertex J leaves the other two vertices, which form side KL.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-opposite-side-0004/)

Triangle · qf-triangle-opposite-side · candidate





## 80. Side Opposite a Vertex

Triangle MNP has explicitly distinct noncollinear vertices. Which undirected side is opposite vertex N?

View answer and explanationMP

Removing vertex N leaves the other two vertices, which form side MP.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-opposite-side-0005/)

Triangle · qf-triangle-opposite-side · candidate





## 81. Side Opposite a Vertex

Triangle QRS has explicitly distinct noncollinear vertices. Which undirected side is opposite vertex S?

View answer and explanationQR

Removing vertex S leaves the other two vertices, which form side QR.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-opposite-side-0006/)

Triangle · qf-triangle-opposite-side · candidate





## 82. Side Opposite a Vertex

Triangle TUV has explicitly distinct noncollinear vertices. Which undirected side is opposite vertex T?

View answer and explanationUV

Removing vertex T leaves the other two vertices, which form side UV.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-opposite-side-0007/)

Triangle · qf-triangle-opposite-side · candidate





## 83. Side Opposite a Vertex

Triangle WXY has explicitly distinct noncollinear vertices. Which undirected side is opposite vertex X?

View answer and explanationWY

Removing vertex X leaves the other two vertices, which form side WY.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-opposite-side-0008/)

Triangle · qf-triangle-opposite-side · candidate





## 84. Side Opposite a Vertex

Triangle ABD has explicitly distinct noncollinear vertices. Which undirected side is opposite vertex D?

View answer and explanationAB

Removing vertex D leaves the other two vertices, which form side AB.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-opposite-side-0009/)

Triangle · qf-triangle-opposite-side · candidate





## 85. Side Opposite a Vertex

Triangle ACE has explicitly distinct noncollinear vertices. Which undirected side is opposite vertex A?

View answer and explanationCE

Removing vertex A leaves the other two vertices, which form side CE.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-opposite-side-0010/)

Triangle · qf-triangle-opposite-side · candidate





## 86. Side Opposite a Vertex

Triangle BDF has explicitly distinct noncollinear vertices. Which undirected side is opposite vertex D?

View answer and explanationBF

Removing vertex D leaves the other two vertices, which form side BF.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-opposite-side-0011/)

Triangle · qf-triangle-opposite-side · candidate





## 87. Side Opposite a Vertex

Triangle CEG has explicitly distinct noncollinear vertices. Which undirected side is opposite vertex G?

View answer and explanationCE

Removing vertex G leaves the other two vertices, which form side CE.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-opposite-side-0012/)

Triangle · qf-triangle-opposite-side · candidate





## 88. Side Opposite a Vertex

Triangle DFH has explicitly distinct noncollinear vertices. Which undirected side is opposite vertex D?

View answer and explanationFH

Removing vertex D leaves the other two vertices, which form side FH.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-opposite-side-0013/)

Triangle · qf-triangle-opposite-side · candidate





## 89. Side Opposite a Vertex

Triangle EGI has explicitly distinct noncollinear vertices. Which undirected side is opposite vertex G?

View answer and explanationEI

Removing vertex G leaves the other two vertices, which form side EI.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-opposite-side-0014/)

Triangle · qf-triangle-opposite-side · candidate





## 90. Side Opposite a Vertex

Triangle FHJ has explicitly distinct noncollinear vertices. Which undirected side is opposite vertex J?

View answer and explanationFH

Removing vertex J leaves the other two vertices, which form side FH.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-opposite-side-0015/)

Triangle · qf-triangle-opposite-side · candidate





## 91. Side Opposite a Vertex

Triangle GIK has explicitly distinct noncollinear vertices. Which undirected side is opposite vertex G?

View answer and explanationIK

Removing vertex G leaves the other two vertices, which form side IK.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-opposite-side-0016/)

Triangle · qf-triangle-opposite-side · candidate





## 92. Side Opposite a Vertex

Triangle HJL has explicitly distinct noncollinear vertices. Which undirected side is opposite vertex J?

View answer and explanationHL

Removing vertex J leaves the other two vertices, which form side HL.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-opposite-side-0017/)

Triangle · qf-triangle-opposite-side · candidate





## 93. Side Opposite a Vertex

Triangle IKM has explicitly distinct noncollinear vertices. Which undirected side is opposite vertex M?

View answer and explanationIK

Removing vertex M leaves the other two vertices, which form side IK.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-opposite-side-0018/)

Triangle · qf-triangle-opposite-side · candidate





## 94. Side Opposite a Vertex

Triangle JLN has explicitly distinct noncollinear vertices. Which undirected side is opposite vertex J?

View answer and explanationLN

Removing vertex J leaves the other two vertices, which form side LN.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-opposite-side-0019/)

Triangle · qf-triangle-opposite-side · candidate





## 95. Side Opposite a Vertex

Triangle KMO has explicitly distinct noncollinear vertices. Which undirected side is opposite vertex M?

View answer and explanationKO

Removing vertex M leaves the other two vertices, which form side KO.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-opposite-side-0020/)

Triangle · qf-triangle-opposite-side · candidate





## 96. Side Opposite a Vertex

Triangle LNP has explicitly distinct noncollinear vertices. Which undirected side is opposite vertex P?

View answer and explanationLN

Removing vertex P leaves the other two vertices, which form side LN.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-opposite-side-0021/)

Triangle · qf-triangle-opposite-side · candidate





## 97. Side Opposite a Vertex

Triangle MOQ has explicitly distinct noncollinear vertices. Which undirected side is opposite vertex M?

View answer and explanationOQ

Removing vertex M leaves the other two vertices, which form side OQ.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-opposite-side-0022/)

Triangle · qf-triangle-opposite-side · candidate





## 98. Side Opposite a Vertex

Triangle NPR has explicitly distinct noncollinear vertices. Which undirected side is opposite vertex P?

View answer and explanationNR

Removing vertex P leaves the other two vertices, which form side NR.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-opposite-side-0023/)

Triangle · qf-triangle-opposite-side · candidate





## 99. Side Opposite a Vertex

Triangle OQS has explicitly distinct noncollinear vertices. Which undirected side is opposite vertex S?

View answer and explanationOQ

Removing vertex S leaves the other two vertices, which form side OQ.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-opposite-side-0024/)

Triangle · qf-triangle-opposite-side · candidate





## 100. Side Opposite a Vertex

Triangle PRT has explicitly distinct noncollinear vertices. Which undirected side is opposite vertex P?

View answer and explanationRT

Removing vertex P leaves the other two vertices, which form side RT.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260925-triangle-opposite-side-0025/)

Triangle · qf-triangle-opposite-side · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20260925-triangle-object-language/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20260925-triangle-object-language.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20260925-triangle-object-language/validation-report.json)