Two circles touch externally at a point P. From a point T on the tangent at P, tangents TQ and TR are drawn to the circles with points of contact Q and R respectively. Prove that TQ = TR

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 .

In Fig 10.57, a circle is inscribed in a quadrilateral ABCD in which . If AD = 23 cm, AB = 29 cm and DS = 5 cm, find the radius r of the circle.

In Fig. 10.58, there are two concentric circles with centre O of radii 5 cm and 3 cm. from an external point P, tangents PA and PB are drawn to these circles. If AP = 12 cm, find the length of BP.