Question

prove that the perpendicular bisector of the sides of cyclic quadrilateral are concurrent

Answer 1

Let ABCD be the cyclic quadrilateral.
Let OL, OM, ON and OP be the perpendicular bisectors of sides AB, BC, CD and AD respectively.
Recall that perpendicular bisector of a line segment passes through the centre of the circle.
Hence OL, OM, ON and OP pass through the centre.
⇒ OL, OM, ON and OP are concurrent.
Thus the perpendicular bisectors of sides of cyclic quadrilateral are concurrent.