Question

prove that if two circles are tangent externally then the common internal tangent bisects a common external tangent.

Answer 1

Explanation:

Consider the attached diagram.

Circles A and B have mutual tangents CF and DE that intersect at point F.

We know that the two tangents to a circle from an external point are congruent, so ...

  FE≅ FC

  FD ≅ FC

By the transitive property of congruence, both segments congruent to FC are congruent to each other:

  FD ≅ FE

Therefore, CF is a bisector of DE.