In Fig., common tangents PQ and RS to two circles intersect at A. Prove that PQ = RS.

Given: PQ and RS are the two-common tangent to the two circles.

To Proof: A is the point of intersection of PQ and RS.

Theorem Used: The length of two tangents drawn from an externa point are equal

Explanation:

We know that, length of two tangent drawn from an exterior point to circle are equal.

Therefore

As A is external point and AP and AR are tangents,

By theorem stated,

PA = RA ------- (i)

As A is external point and AS and AQ are tangents,

QA = SA --------- (ii)

Adding two equations we get

PA + QA =RA + SA

PQ =RS (proved)