Question

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

Answer 1

Given: Two circles with centre's $O_1$ and $O_2$. AB and CD are common tangents to the circles which intersect in P.
To Prove: AB = CD
Proof:
AP = PC ..... (1) (length of tangents drawn from an external point to the circle are equal)
PB = PD ...... (2) (length of tangents drawn from an external point to the circle are equal)
Adding (1) and (2), we get
AP + PB = PC + PD
$\Rightarrow$ AB = CD