In Figure 1, common tangents AB and CD to the two circles with centres O₁ and O₂ intersect at E. Prove that AB = CD.

The incircle of an isosceles triangle ABC, in which AB = AC, touches the sides BC, CA and AB at D, E and F respectively. Prove that BD = DC.

Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.

Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

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If the angle between two tangents drawn from an external point P to a circle of radius a and centre O, is 60°, then find the length of OP.

A circle touches all the four sides of a quadrilateral ABCD. Prove that 

AB + CD = BC + DA

Prove that the tangents drawn at the end points of a chord of a circle make equal angles with the chord.

In the given figure, XY and X’Y’ are two parallel tangents to a circle with centre O and another tangents AB with point of contact C, is intersecting XY at A and X’Y’ at B. Prove that ∠AOB = 90°.

Prove that the lengths of two tangents drawn from an external point to a circle are equal.

In Fig., the sides AB, BC and CA of a triangle ABC, touch a circle at P, Q and R respectively. If PA = 4 cm, BP = 3 cm and AC = 11 cm, then the length of BC (in cm) is

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