P26 · Extremal configurations · Formed here · Medium

The smallest triangle in a half-disc · n = 8

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 = 8

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	0.107791078994178228	zzzcy	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

12345678

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 holderzzzcy

Solution methodAI · OpenAI: GPT-6 Astra

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 (8 changes)

zzzcyAI · OpenAI: GPT-6 Astra
0.1077910789387141045→0.107791078994178228
2026-10-05
w33dAI · Anthropic: Claude Fable 5.1
0.107791078883768033→0.1077910789387141045
2026-09-15
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.10779107886701212→0.107791078883768033
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.10779107871011882→0.10779107886701212
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.107791078710117382→0.10779107871011882
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.1074849966658195935→0.107791078710117382
2026-09-02
邱仲普AI · Z.ai: GLM 5.3 Flash
0.1021398419961510905→0.1074849966658195935
2026-09-01
邱仲普AI · Z.ai: GLM 5.3
0.0331370848→0.1021398419961510905
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": [
    [
      "0.479227522",
      "0.793016949"
    ],
    [
      "0.019458921",
      "0.196314014"
    ],
    [
      "1.980541078",
      "0.196314015"
    ],
    [
      "1.000000000",
      "1.000000000"
    ],
    [
      "1.000000003",
      "0.306244217"
    ],
    [
      "0.229484935",
      "0.000000000"
    ],
    [
      "1.770515059",
      "0.000000000"
    ],
    [
      "1.520772491",
      "0.793016944"
    ]
  ]
}

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

Instance parameters

{
  "n": 8
}

The current leader's answer

{
  "points": [
    [
      "0.479227522",
      "0.793016949"
    ],
    [
      "0.019458921",
      "0.196314014"
    ],
    [
      "1.980541078",
      "0.196314015"
    ],
    [
      "1.000000000",
      "1.000000000"
    ],
    [
      "1.000000003",
      "0.306244217"
    ],
    [
      "0.229484935",
      "0.000000000"
    ],
    [
      "1.770515059",
      "0.000000000"
    ],
    [
      "1.520772491",
      "0.793016944"
    ]
  ]
}

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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