Prove that the perpendicular bisectors of the sides of a cyclic quadrilateral are concurrent.

Let ABCD be a cyclic quadrilateral, and let O be the center of the corresponding circle Then, each side of the equilateral ABCD is a chord of the circle and the perpendicular bisector of a chord always passes through the center of the circle 

So, right bisectors of the sides of quadrilaterals ABCD, will pass through the circle O of the corresponding circle

Let ABCD be a cyclic quadrilateral and let O be the centre of the corresponding circle 

Then, 

Each side of the equilateral ABCD is a chord of the circle and the perpendicular bisector of a chord always passes through the centre of the circle 

So, 

Right bisectors of the sides of the quadrilateral ABCD will pass through the centre O of the corresponding circle.