Proposition: Prop. 12.01: Areas of Similar Polygons Inscribed in Circles are as Squares on Diameters

Euclid's Formulation

Similar polygons (inscribed) in circles are to one another as the squares on the diameters (of the circles). * Let $ABC$ and $FGH$ be circles, and let $ABCDE$ and $FGHKL$ be similar polygons (inscribed) in them (respectively), and let $BM$ and $GN$ be the diameters of the circles (respectively). * I say that as the square on $BM$ is to the square on $GN$, so polygon $ABCDE$ (is) to polygon $FGHKL$.


Modern Formulation

(not yet contributed)

Proofs: 1

Proofs: 1 2

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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