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.


Modern Formulation

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

Proofs: 1

Thank you to the contributors under CC BY-SA 4.0!



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