Proposition: 3.03: Conditions for Diameter to be a Perpendicular Bisector

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

In a circle, if any straight line through the center cuts in half any straight line not through the center then it also cuts it at right angles. And (conversely) if it cuts it at right angles then it also cuts it in half. * Let $ABC$ be a circle, and, within it, let some straight line through the center, $CD$, cut in half some straight line not through the center, $AB$, at the point $F$. * I say that ($CD$) also cuts ($AB$) at right angles. fig03e

Modern Formulation

Let $AB$ be a chord, which does not go through the center $E$ of a given circle and let $CD$ be its diameter, i.e. it goes through the center $E.$

Then $CD$ bisects $AB$ if and only if $CD\perp AB,$ i.e. both are perpendicular.

Proofs: 1

Proofs: 1 2 3 4 5

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