In the figure (i) given below, two circles intersect at A, B. From a point P on one of these circles, two line segments PAC and PBD are drawn, intersecting the other circles at C and D respectively. Prove that CD is parallel to the tangent at P.

In the figure (ii) given below, two circles with centres C, C' intersect at A, B and the point C lies on the circle with C'. PQ is a tangent to the circle with centre C' at A. Prove that AC bisects ∠PAB.

If the sides of a rectangle touch a circle, prove that the rectangle is a square.

In the figure (ii) given below, O and O' are centres of two circles touching each other externally at the point P. The common tangent at P meets a direct common tangent AB at M. Prove that,

(i) M bisects AB.

(ii) ∠APB = 90°.