Proposition: 3.09: Condition for Point to be Center of Circle

(Proposition 9 from Book 3 of Euclid's “Elements”)

If some point is taken inside a circle, and more than two equal straight lines radiate from the point towards the (circumference of the) circle, then the point taken is the center of the circle. * Let $ABC$ be a circle, and $D$ a point inside it, and let more than two equal straight lines, $DA$, $DB$, and $DC$, radiate from $D$ towards (the circumference of) circle $ABC$. * I say that point $D$ is the center of circle $ABC$.


Modern Formulation

If a point $D$ can be connected with points on a circumference of a given circle such that more than two connecting segments have the same length, then $D$ must be 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