Proposition: Prop. 13.12: Square on Side of Equilateral Triangle inscribed in Circle is Triple Square on Radius of Circle

(Proposition 12 from Book 13 of Euclid's “Elements”)

If an equilateral triangle is inscribed in a circle then the square on the side of the triangle is three times the (square) on the radius of the circle. * Let there be a circle $ABC$, and let the equilateral triangle $ABC$ have been inscribed in it [Prop. 4.2]. * I say that the square on one side of triangle $ABC$ is three times the (square) on the radius of circle $ABC$.


Modern Formulation

(not yet contributed)

Proofs: 1

Proofs: 1

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Adapted from (subject to copyright, with kind permission)

  1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"

Adapted from CC BY-SA 3.0 Sources:

  1. Prime.mover and others: "Pr∞fWiki",, 2016