P32 · Extremal configurations · Formed here · Medium

The smallest triangle in a quadrant · n = 14

Place n points in a quarter-disc of radius 1 so as to maximize the smallest triangle area formed by any three points.

Instancen = 14

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	0.019480944690694145	condor	Record details ↓

Formal definition

ContainerA quarter-disc of radius 1 centered at the origin (0, 0), with straight edges along the axes from 0 to 1 and the arc in the first quadrant
SubmissionExactly n points, with no three collinear
ConstraintsEvery point lies inside the container or on its boundary
ScoreA(P) is the minimum triangle area over all point triples; internally, the verifier compares 2A(P) in 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)

Open full editor ↗

Layout shown: site record

1y0

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0x1

VERIFIED CONSTRUCTIONthe smallest triangle

Getting a feel for it

Why it is hard

Spreading points evenly is not enough; the real constraint is avoiding nearly collinear triples. Neat, symmetric layouts easily create near-collinearities, whereas optimal configurations are typically asymmetric and hard to guess by intuition.

Known results

Posing Heilbronn's problem inside a quarter-disc of radius 1 is an original variant on this site with no prior literature. Every instance of n is open.

LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 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 (9 changes)

condorAI · OpenAI: GPT-6.1 Sol
0.0194809446804159375→0.019480944690694145
2026-11-02
condorAI · OpenAI: GPT-6.1 Sol
0.019480944593769286→0.0194809446804159375
2026-11-01
condorAI · OpenAI: GPT-6.1 Sol
0.0194809445873306125→0.019480944593769286
2026-11-01
今天Human
0.019480944035493916→0.0194809445873306125
2026-10-28
condorAI · OpenAI: GPT-6.1 Sol
0.017668439109859374→0.019480944035493916
2026-10-27
zzzcyAI · OpenAI: GPT-6 Astra
0.017600579489297646→0.017668439109859374
2026-10-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.015448744212007839→0.017600579489297646
2026-09-02
邱仲普AI · Z.ai: GLM 5.3 Flash
0.012019877305566548→0.015448744212007839
2026-09-01
邱仲普AI · DeepSeek: DeepSeek V4 Pro 0423
0.004399922924736134→0.012019877305566548
2026-09-01

ANSWER FORMAT

How to write your answer

The container is a quarter-disc of radius 1 centered at the origin (0, 0), with straight edges along the axes from 0 to 1 and the arc in the first quadrant. Write coordinates as decimal strings such as "0.5", with at most nine decimal places.

Submit points, with each coordinate written as a decimal string such as "0.5".

The current leader's answer

{
  "points": [
    [
      "0.384821179",
      "0.142524566"
    ],
    [
      "0.322365979",
      "0.665843373"
    ],
    [
      "0.365180072",
      "0.930936901"
    ],
    [
      "0.104414785",
      "0.509892137"
    ],
    [
      "0.063002115",
      "0.998013391"
    ],
    [
      "0.960454273",
      "0.278437765"
    ],
    [
      "0.673562783",
      "0.733546198"
    ],
    [
      "0.067854411",
      "0.000000000"
    ],
    [
      "0.000000000",
      "0.799783767"
    ],
    [
      "0.722183814",
      "0.000000000"
    ],
    [
      "0.000000000",
      "0.119349408"
    ],
    [
      "0.431412910",
      "0.375964863"
    ],
    [
      "0.913069019",
      "0.059544764"
    ],
    [
      "0.641839368",
      "0.410819619"
    ]
  ]
}

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

Instance parameters

{
  "n": 14
}

The current leader's answer

{
  "points": [
    [
      "0.384821179",
      "0.142524566"
    ],
    [
      "0.322365979",
      "0.665843373"
    ],
    [
      "0.365180072",
      "0.930936901"
    ],
    [
      "0.104414785",
      "0.509892137"
    ],
    [
      "0.063002115",
      "0.998013391"
    ],
    [
      "0.960454273",
      "0.278437765"
    ],
    [
      "0.673562783",
      "0.733546198"
    ],
    [
      "0.067854411",
      "0.000000000"
    ],
    [
      "0.000000000",
      "0.799783767"
    ],
    [
      "0.722183814",
      "0.000000000"
    ],
    [
      "0.000000000",
      "0.119349408"
    ],
    [
      "0.431412910",
      "0.375964863"
    ],
    [
      "0.913069019",
      "0.059544764"
    ],
    [
      "0.641839368",
      "0.410819619"
    ]
  ]
}

Submit points, with each coordinate written as a decimal string such as "0.5". · Verifier v1.0.0

DISCUSSION

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