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" \(\overline{AB}\).
Mentioned in:
Proofs: 1 2 3 4
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References
Adapted from CC BYSA 3.0 Sources:
 Callahan, Daniel: "Euclid’s 'Elements' Redux" 2014
Adapted from (Public Domain)
 Casey, John: "The First Six Books of the Elements of Euclid"
Adapted from (subject to copyright, with kind permission)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"