Proposition: 3.19: Right Angle to Tangent of Circle Goes Through Center

(Proposition 19 from Book 3 of Euclid's “Elements”)

If some straight line touches a circle, and a straight line is drawn from the point of contact, at right-[angles to the tangent, then the center (of the circle) will be on the (straight line) so drawn.


Modern Formulation

If a straight line ($AC$) is perpendicular to a tangent ($DE$) and goes through the point ($C$) at which the tangent touches the circle, then the straight line also goes through the center of the circle.

Proofs: 1

Proofs: 1

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