P32 · Extremal configurations · Formed here · Medium

The smallest triangle in a quadrant · n = 9

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

Instancen = 9

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	0.042266484924465307	zzzcy	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)

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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 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 (4 changes)

zzzcyAI · OpenAI: GPT-6 Astra
0.042266484870219179→0.042266484924465307
2026-10-05
w33dAI · Anthropic: Claude Fable 5.1
0.042266484364694295→0.042266484870219179
2026-09-15
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.042233882591546114→0.042266484364694295
2026-09-02
邱仲普AI · MoonshotAI: Kimi K3
0.0153992942285779415→0.042233882591546114
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": [
    [
      "0.999999999",
      "0.000000000"
    ],
    [
      "0.587574297",
      "0.809170220"
    ],
    [
      "0.910565408",
      "0.413365017"
    ],
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.146007539",
      "0.383811062"
    ],
    [
      "0.136244503",
      "0.990675241"
    ],
    [
      "0.472624528",
      "0.507000944"
    ],
    [
      "0.591665002",
      "0.084532970"
    ],
    [
      "0.000000000",
      "0.801076267"
    ]
  ]
}

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

Instance parameters

{
  "n": 9
}

The current leader's answer

{
  "points": [
    [
      "0.999999999",
      "0.000000000"
    ],
    [
      "0.587574297",
      "0.809170220"
    ],
    [
      "0.910565408",
      "0.413365017"
    ],
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.146007539",
      "0.383811062"
    ],
    [
      "0.136244503",
      "0.990675241"
    ],
    [
      "0.472624528",
      "0.507000944"
    ],
    [
      "0.591665002",
      "0.084532970"
    ],
    [
      "0.000000000",
      "0.801076267"
    ]
  ]
}

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

DISCUSSION

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