Inscribe the largest equilateral triangle

Drop three points on the shape’s outline so they form an equilateral triangle — all three sides the same length, all three angles 60° — and make it as large as possible. The puzzle scores you on both regularity and size relative to the maximum equilateral triangle with vertices on the shape’s outline. Same placement mechanics as the square variation: the nearest boundary point follows your cursor, tap to drop, drag to refine, confirm to score.

The math: Nielsen–Wright and friends

Unlike the Inscribed Square Problem, the equilateral-triangle case is fully solved. In 1990, Mark Nielsen and Stephanie Wright proved that every Jordan curve contains inscribed equilateral triangles — in fact, infinitely many of them. Simpler proofs exist for smooth and polygonal curves using the same IVT-style continuity tricks that drive the cut-in-half puzzle.

The sketch: pick any point A on the curve. For every direction θ you can construct a candidate equilateral triangle with A as one vertex and a second vertex lying on the curve at distance r(θ) in direction θ. As θ rotates, the third vertex traces a continuous curve that must intersect the original outline — and at every intersection, you get an inscribed equilateral triangle.

Tips

  • Start with two points a reasonable distance apart — they define one side of the triangle.
  • The third vertex is the apex of an equilateral triangle built on that side. There are two possible apex positions (one on each side). Pick the one that actually lands on the outline.
  • Long narrow shapes have smaller inscribed equilateral triangles than you’d expect.
  • If one vertex is locked on a sharp corner of the shape, try a second vertex on the opposite long edge — triangles often snap nicely that way.

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