Draw any chord in the circle
Construct a perpendicular bisector to this chord.....the center will lie on this bisector.....so, at both points where the bisector intersects the circle, we have a diameter
Construct a perpendicular bisector to this diameter........the center of the circle lies at the point of bisection
Here's a proof...with a picture : https://mathcs.clarku.edu/~djoyce/java/elements/bookIII/propIII1.html