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In Figure 1, common tangents AB and CD to the two circles with centres 0_{1}and 0_{2 }intersect at E. Prove that AB = CD.

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#### Solution

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.

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