Proposition: 3.18: Radius at Right Angle to Tangent

(Proposition 18 from Book 3 of Euclid's “Elements”)

If some straight line touches a circle, and some (other) straight line is joined from the center (of the circle) to the point of contact, then the (straight line) so joined will be perpendicular to the tangent. * For let some straight line $DE$ touch the circle $ABC$ at point $C$, and let the center $F$ of circle $ABC$ have been found [Prop. 3.1], and let $FC$ have been joined from $F$ to $C$. * I say that $FC$ is perpendicular to $DE$.


Modern Formulation

If a tangent and a radius of a given circle intersect, then they are perpendicular.

Proofs: 1

Proofs: 1 2 3 4 5 6

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