Question

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

Answer 1

Given: Circles with centres A and B touches externally. Line l is the tangent to both the circles at P and Q, respectively.

Line m is common tangent which intersects at point R to line l.

To prove: Line m bisects segment PQ i.e., PR = Q

Proof: We know that,

Tangents drawn from an external point to a circle are congruent.

Consider a circle with centre A. Here, point R is an external point and seg RP and seg RC are the tangents.

∴ RP = RC   ...(i) [Tangent Segment Theorem]

Now, consider the circle with centre B. Here, point R is an external point and RQ and RC are tangents.

∴ RQ = RC   ...(ii) [Tangent Segment Theorem]

From equations (i) and (ii), we get

RP = RQ

i.e., line m bisects segment PQ

Hence Proved.