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.


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