Two equal chords AB and CD of a circle with center O, intersect each other at point P inside the circle. Prove that :

(i) AP = CP

(ii) BP = DP

In the following figure, OABC is a square. A circle is drawn with O as center which meets OC at P and OA at Q. Prove that :

(i) △ OPA ≅ △ OQC

(ii) △ BPC ≅ △ BQA

The line joining the mid-points of two chords of a circle passes through its center. Prove that the chords are parallel.

In the following figure, the line ABCD is perpendicular to PQ; where P and Q are the centers of the circles. Show that :

(i) AB = CD

(ii) AC = BD