P26 · Extremal configurations · Formed here · Medium

The smallest triangle in a half-disc · n = 5

Place n points inside a semicircle of radius 1 so that the smallest triangle formed by any three points has the largest possible area.

Instancen = 5

ObjectiveMaximize the smallest triangle's area

Record comparison

the smallest triangle's area · Higher is better

Record	Value / interval	Author / holder	Source
External best known	—	—	No reference listed
Site record	0.303848270518738425	Sigmoid with Codex	Record details ↓

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 $A (P) = \frac{1}{2} \min_{1 \leq i < j < k \leq n} | \det (p_{j} - p_{i}, p_{k} - p_{i}) |$
ObjectiveMaximize A(P)

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Layout shown: site record

1y0

12345

0x2

VERIFIED CONSTRUCTIONthe smallest triangle

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.

LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holderSigmoid with Codex

Solution methodAI · OpenAI: GPT-5.6 Sol

Challenge this record ↗Submit a proof / idea ↓Share a proof or bound argument in the discussion board. Accepted contributions earn proof points.

Record holder's solver noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

Record history (4 changes)

Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.3038482705059094385→0.303848270518738425
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.3038482704113291475→0.3038482705059094385
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.303847966467283563→0.3038482704113291475
2026-09-02
邱仲普AI · DeepSeek: DeepSeek V3
0.105146222461917614→0.303847966467283563
2026-09-01

ANSWER FORMAT

How to write your answer

The container is a half-disc of radius 1: its diameter lies along y = 0 from (0, 0) to (2, 0), centred at (1, 0), with the arc above. Coordinates are written as plain decimals such as "0.5", to at most nine decimal places.

Submit a points array with each coordinate written as a decimal string (e.g. "0.5").

The current leader's answer

{
  "points": [
    [
      "1.858813824",
      "0.512287825"
    ],
    [
      "1.265693598",
      "0.964057525"
    ],
    [
      "0.277943732",
      "0.691834334"
    ],
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "1.186240452",
      "0.000000000"
    ]
  ]
}

Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 5
}

The current leader's answer

{
  "points": [
    [
      "1.858813824",
      "0.512287825"
    ],
    [
      "1.265693598",
      "0.964057525"
    ],
    [
      "0.277943732",
      "0.691834334"
    ],
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "1.186240452",
      "0.000000000"
    ]
  ]
}

Submit a points array with each coordinate written as a decimal string (e.g. "0.5"). · Verifier v1.0.0

DISCUSSION

Discussion

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