Question

In the given figure, two circles touch each other at the point C. Prove that the common tangent to the circles at C, bisects the common tangent at P and Q.

Answer 1

Given: Two circles with centres A and B touch each other externally at point C.

PQ is a common external tangent touching the circles at P and Q.

The common tangent at point C meets PQ at point T.

To prove: T bisects PQ, i.e., PT = TQ.

Proof:

Since, PT = TC   ...(Tangents of circle)

And QT = TC   ...(Tangents of circle from extended point)

So, PT = QT

Now, PQ = PT + TQ

⇒ PQ = PT + PT

⇒ PQ = 2PT

⇒ `1/2` PQ = PT

Hence, the common tangent to the circle at C, bisects the common tangents at P and Q.