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Proposition: 6.31: Similar Figures on Sides of Right-Angled Triangle

(Proposition 31 from Book 6 of Euclid's “Elements”)

In right-angled triangles, the figure (drawn) on the side subtending the right angle is equal to the (sum of the) similar, and similarly described, figures on the sides surrounding the right angle.

• Let $ABC$ be a right-angled triangle having the angle $BAC$ a right angle.
• I say that the figure (drawn) on $BC$ is equal to the (sum of the) similar, and similarly described, figures on $BA$ and $AC$.

Modern Formulation

This is a generalization of the Pythagorean theorem, in which the squares are replaced by similar rectangles.

Table of Contents

Proofs: 1

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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