The smallest triangle in a half-disc
Place n points inside a semicircle of radius 1 so that the smallest triangle formed by any three points has the largest possible area.
Formal definition
- ContainerAn upper semicircle of radius 1: the diameter runs along y = 0 from (0, 0) to (2, 0), centered at (1, 0), with the circular arc in y ≥ 0
- SubmissionExactly n points, with no three collinear
- ConstraintsEvery point must lie inside the container or on its boundary
- ScoreA(P) is the minimum triangle area over all triples of points; internally, the verifier compares 2A(P) using exact arithmetic
- ObjectiveMaximize A(P)
Getting a feel for it
How to push the minimum area up
Simply spreading the points out evenly is not enough: whenever three points are nearly collinear, the triangle they form collapses into a thin sliver. A semicircle combines a curved arc with a flat diameter base, and at most two points can sit on the diameter without becoming collinear, so using the curvature of the upper arc to stagger every line is key.
Current status
Heilbronn's triangle problem inside a semicircle of radius 1 is an original variant introduced on this site, with no prior published literature. Every n is unexplored territory, and the current leaderboard records are the best results known.
Current best solutions by n
Each parameter n is an independently scored sub-problem. Select any card to inspect the current record construction or submit a stronger solution.
Discussion (0) ↓Data and citation
All sub-problems in this family — authoritative scores, proof status, coordinates, and sources — are published at the stable URL below under CC BY 4.0. Because scores update as records fall, please cite the file's generatedAt timestamp.
GET https://minmaxarena.com/data/heilbronn-in-a-semicircle.json
Cite the frozen 2026-10 edition: live records change, whereas a frozen edition stays fixed and reproducible.
GET https://minmaxarena.com/data/editions/2026-10/heilbronn-in-a-semicircle.json
BibTeX (click to copy)
@misc{minmaxarena-heilbronn-in-a-semicircle-2026-10,
title = {{The smallest triangle in a half-disc} (P26)},
author = {{MinMax Arena}},
year = {2026},
note = {Machine-verified records, 2026-10 edition},
url = {https://minmaxarena.com/data/editions/2026-10/heilbronn-in-a-semicircle.json},
license = {CC BY 4.0}
}Discussion
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