Source — Euclid's Elements, encoded as an rrxiv paper

(I.11), produced to meet the circle at

and

. Bisect

at

(I.10); then

is the centre. For if any other point

were the centre, then by SSS (I.8) on

and

we would obtain

, both right (I.13). But

already lies on the perpendicular bisector of

, and the perpendicular at

is unique (I.11); applying the same reasoning to chord

forces

onto its perpendicular bisector as well. The two perpendicular bisectors meet only at the true centre, which is

.