Proposition: 3.32: Angles made by Chord with Tangent

(Proposition 32 from Book 3 of Euclid's “Elements”)

If some straight line touches a circle, and some (other) straight line is drawn across, from the point of contact into the circle, cutting the circle (in two), then those angles the (straight line) makes with the tangent will be equal to the angles in the alternate segments of the circle.


Modern Formulation

In a circle, let $\overline{BA}$ be its diameter and $\boxdot{ABCD}$ be a quadrilateral inscribed in the circle and let a tangent $EF$ touch the circle at the point $B$ and let the straight line $BD$ cut the circle (see figure). Then following rectilinear angles are equal: $$\angle{FBD}=\angle{BAD},\quad \angle{DBE}=\angle{DCB}.$$

Proofs: 1

Proofs: 1 2 3

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