Proof of a certain lemma in geometry

Question

In the following article: http://yufeizhao.com/olympiad/geolemmas.pdf in the proof of the fact about the diameter of the incircle on page 2, the author claims that the proof that $BD = CF$ follows easily from the fact that $F$ is the tangency point of the excircle corresponding to $A$ with $BC$ (see the article for the definition of the labels in the problem). However, I don't see why this is true. What is the proof of this fact?

Answer 1

Since tangent segments have equal length, we have $AB-BD=AC-CD$ and $AB+BF=AC+CF$, and of course we have $BD+CD=BF+CF$. Hence $$ (AC+CF)+(AB-BD)+(BF+CF)=(AB+BF)+(AC-CD)+(BD+CD)$$ which simplifies to $BD=CF$.