Proposition: Prop. 13.04: Area of Squares on Whole and Lesser Segment of Straight Line cut in Extreme and Mean Ratio

(Proposition 4 from Book 13 of Euclid's “Elements”)

If a straight line is cut in extreme and mean ratio then the sum of the squares on the whole and the lesser piece is three times the square on the greater piece. * Let $AB$ be a straight line, and let it have been cut in extreme and mean ratio at $C$, and let $AC$ be the greater piece. * I say that the (sum of the squares) on $AB$ and $BC$ is three times the (square) on $CA$.


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