Question

In figure, two circles touch each other at point C. prove that the common tangent to the Circles at C, bisects the common tangent at p and q

Answer 1

Let PQ and RS are two direct common tangents and EF is the transverse common tangent.

We know that tangent line segments are equal in length from an external point to circle.

Therefore, EP = EC and EQ = EC ⇒ EP = EQ

and FR = FC, FS = FC ⇒ FR = FS

Hence, ECF bisects PQ and ECF bisects RS.

∴ The common tangent bisects the other two common tangents

Answer 2

Solution :

Given,
two circle touch each other at point C. RC and PQ are two common tangent.

We have to prove that,
CR bisects PQ
means, PR=RQ

Proof :
We know that,
two tangents from a common point are same
So, PR=CR ------(1)
and QR=CR-------(2)

From (1)&(2)
PR=QR
.
. . CR bisectst PQ