Proposition: 6.08: Perpendicular in Right-Angled Triangle makes two Similar Triangles

(Proposition 8 from Book 6 of Euclid's “Elements”)

If, in a right-angled triangle, a (straight line) is drawn from the right angle perpendicular to the base then the triangles around the perpendicular are similar to the whole (triangle), and to one another.


Modern Formulation

If in a right-angled triangle a straight line is drawn from the vertex of the right angle to the hypotenuse, then the resulting two triangles are similar.

Proofs: 1 Corollaries: 1

Proofs: 1 2 3 4 5 6 7 8 9 10

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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