Axiom: 1.3: Circle Determined by its Center and its Radius

(Postulate 3 from Book 1 of Euclid's “Elements”)

And to draw a circle with any center":bookofproofs$690 and "radius":bookofproofs$690.

Modern Formulation

A point \(A\) and a given segment \(\overline{AB}\) always completely determine a circle \(c\). Instead of saying "determine", we also say that \(c\) "is the circle with the center" \(A\) "of the radius" \(\overline{AB}\).

Proofs: 1 2 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 (Public Domain)

  1. Casey, John: "The First Six Books of the Elements of Euclid"

Adapted from (subject to copyright, with kind permission)

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