Definition: 3.04: Chords Equally Far From the Center of a Circle

In a circle, straight lines are said to be equally far from the center when the perpendiculars drawn to them from the center are equal.

Modern Formulation

Chords are said to be equally distant from the center (or equally far from the center), if the perpendiculars constructed to them from the center are equal in length, i.e. congruent.


In the following figure, the chords \(\overline{AB}\) and \(\overline{CD}\) are equally distant from the center, because the segments perpendicular to the chords have the same length \(a\) (i.e. they are congruent):


Proofs: 1 2
Propositions: 3 4

Thank you to the contributors under CC BY-SA 4.0!



Adapted from CC BY-SA 3.0 Sources:

  1. Callahan, Daniel: "Euclid’s 'Elements' Redux" 2014

Adapted from (subject to copyright, with kind permission)

  1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"