In the figure (i) given below, AD is a diameter of a circle with center O. If AB || CD, prove that AB = CD.

In the figure (ii) given below, AB and CD are equal chords of a circle with center O. If AB and CD meet at E (outside the circle) prove that

(i) AE = CE

(ii) BE = DE.

A and B are points on a circle with center O. C is a point on the circle such that OC bisects ∠AOB, prove that OC bisects the arc AB.

Prove that the angle subtended at the center of a circle is bisected by the radius passing through the mid-point of arc.