Proposition: Prop. 13.03: Area of Square on Lesser Segment of Straight Line cut in Extreme and Mean Ratio

(Proposition 3 from Book 13 of Euclid's “Elements”)

If a straight line is cut in extreme and mean ratio then the square on the lesser piece added to half of the greater piece is five times the square on half of the greater piece. * For let some straight line $AB$ have been cut in extreme and mean ratio at point $C$. * And let $AC$ be the greater piece. * And let $AC$ have been cut in half at $D$. * I say that the (square) on $BD$ is five times the (square) on $DC$.


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