Proposition: 3.20: Inscribed Angle Theorem

(Proposition 20 from Book 3 of Euclid's “Elements”)

In a circle, the angle at the center is double that at the circumference, when the angles have the same circumference base. * Let $ABC$ be a circle, and let $BEC$ be an angle at its center, and $BAC$ (one) at (its) circumference. * And let them have the same circumference base $BC$. * I say that angle $BEC$ is double (angle) $BAC$.


Modern Formulation

The central angle is double the inscribed angle.

Proofs: 1

Proofs: 1 2 3
Sections: 4

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