MathNet 2017-part-02 — 176 problems (EN–VI) Question 105

Let $ABC$ be an isosceles triangle with apex $A$ and altitude $AD$. On $AB$, choose a point $F$ distinct from $B$ such that $CF$ is tangent to the incircle of $ABD$. Suppose that $\triangle BCF$ is isosceles. Show that those conditions uniquely determine:

a) which vertex of $BCF$ is its apex;

b) the size of $\angle BAC$.