Question

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

Answer 1

We know that tangent segments to a circle from the same external point are congruent.
So, we have
EA = EC for the circle having center O₁
and
ED = EB for the circle having center O₁ 
Now, Adding ED on both sides in EA = EC. we get
EA+ ED  =  EC + ED
⇒  EA + EB = EC + ED
⇒  AB = CD