Consider ABCD as a cyclic quadrilateral with centre O passing through the points A, B, C, D
We know that AB, BC, CD and DA are the chords of the circle and its right bisector passing through the centre O
So we know that the right bisectors of AB, BC, CD and DA pass through the centre O and are concurrent.
Therefore, it is proved that the perpendicular bisectors of the sides of a cyclic quadrilateral are concurrent.