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. * Let $ABC$ be a right-angled triangle having the angle $BAC$ a right angle, and let $AD$ have been drawn from $A$, perpendicular to $BC$ [Prop. 1.12]. * I say that triangles $ABD$ and $ADC$ are each similar to the whole (triangle) $ABC$ and, further, 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

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