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Proposition: Prop. 13.01: Area of Square on Greater Segment of Straight Line cut in Extreme and Mean Ratio

(Proposition 1 from Book 13 of Euclid's “Elements”)

If a straight line is cut in extreme and mean ratio then the square on the greater piece, added to half of the whole, is five times the square on the half. * For let the straight line $AB$ have been cut in extreme and mean ratio at point $C$, and let $AC$ be the greater piece. * And let the straight line $AD$ have been produced in a straight line with $CA$. * And let $AD$ be made (equal to) half of $AB$. * I say that the (square) on $CD$ is five times the (square) on $DA$.

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Modern Formulation

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Table of Contents

Proofs: 1

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Proofs: 1 2

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References

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", https://proofwiki.org/wiki/Main_Page, 2016