In Fig. 10.54, a circle touches all the four sides of a quadrilateral ABCD with AB = 6 cm, BC = 7 cm and CD = 4 cm. Find AD.

Two tangent segments PA and PB are drawn to a circle with centre O such that . Prove that OP = 2 AP.

If is isosceles with AB = AC and C (O, r) is the incircle of the touching BC at L, prove that L bisect BC.

Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre of the circle.

In fig. 10.55, O is the centre of the circle and BCD is tangent to it at C. Prove that .