Length of tangents of incircle and excircle

Question

Given a triangle $ABC$ , the incircle touches side $BC $ at $D $ and the excircle touches the side $BC$ at $F$ .

Prove that $BF=CD$ .

Can't think of a way to relate the tangents of the incircle and the excircle.

Any tips please?

Answer 1

I will provide a sketch of the proof:

1. Let $a,b,c$ be the lengths of sides $BC,CA,AB$, respectively, and let $s=\frac{a+b+c}{2}$ be the semi-perimeter of the triangle.

2. Show that $CD=s-c$.

3. Show by similar logic that $BF=s-c$ as well.