Question

In Figure 1, common tangents AB and CD to the two circles with centres 0₁and 0₂ intersect at E. Prove that AB = CD.

Answer 1

Given: AB and CD are common tangents to both the circles.
To prove: AB = CD
Proof:
We know that two tangents drawn to a circle for the same exterior point are
equal.
Thus we get
AE = EC (i)
Similarly
ED = EB (ii)
AB = AE + EB (iii)
and
CD = CE + ED (iv)
AB = EC + EB from (i) and (iii)
CD = EC + EB from (ii) and (iv)
Therefore AB = CD
Hence proved.