Question

In the figure, AB and CD are common tangents to two circles of unequal radi. Prove that AB = CD

Answer 1

Construct a line passing through AD

Now, AD and CD are tangents to the circle with centre O from the external point D. 
So, AD = CD    (Tangents drawn from an external point to a circle are equal.)         .....(1)

Also, AB and AD are the tangents to the circle with centre O' from the external point A.
So, AD = AB    (Tangents drawn from an external point to a circle are equal.)          .....(2)

From (1) and (2)

AB = CD

Hence Proved. 

Answer 2

Given: AB and CD are two common tangents to two circles of unequal radii.

To Prove: AB = CD

Construction: Produce AB and CD, to intersect at P.

Proof: Consider the circle with greater radius.

AP = CP  ......[Tangents drawn from an external point to a circle are equal] [1]

Also, Consider the circle with smaller radius.

BP = BD   ......[Tangents drawn from an external point to a circle are equal] [2]

Substract [2] from [1], we get

AP - BP = CP – BD

AB = CD

Hence Proved .