Question

Are the perpendicular bisectors of the sides of a cyclic quadrilateral concurrent? Sketch an example and make a conjecture. What would you need to know to prove your conjecture?

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. Now AB, BC,CD and AD are the chords of the circle. From the property of circle

" perpendicular drawn from the center of the circle to the chord bisect the chord" , it is justified that OL, OM, ON and OP being perpendicular to the respective chords must pass through the centre. ⇒ OL, OM, ON and OP are concurrent. Thus the perpendicular bisectors of sides of cyclic quadrilateral are concurrent.