Question

Two circles touch each other externally at C. Prove that the common tangent at C bisects the other two common tangents.

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

Proved that if Two circles touch each other externally at C then the common tangent at C bisects the other two common tangents.

Tangent Segments Congruence Theorem

Tangent segments to a circle that share the same external endpoint are congruent:

Two tangents to a circle from an external point are equal in Length

Refer to attached figure:

Assume that other two common tangents are FG and HI

Common Tangent at C intersect FG at E and HI at D.

Now DI and DC are Tangents to same Circle from Same external point D

Hence DI = DC
Similarly DH and DC are Tangents to other Circle Hence DH = DC

DI = DC and DH = DC

Hence DI = DH

Hence D is the mid point of HI

Now EF and EC are Tangents to same Circle from Same external point E

Hence EF = FC
Similarly EG and EC are Tangents to other Circle Hence EG = EC

EF = EC and EG = EC

Hence EF = EG

Hence E is the mid point of FG

Hence DE ( Common tangent at C) bisects other two common tangents HI and FG

Hence Proved

Hence, Proved that if Two circles touch each other externally at C then the common tangent at C bisects the other two common tangents.