Balti Tee 2018 · Ülesanne 14

Denote by $X$ the intersection point of the lines $A_1A_2$ and $AC$. Prove that $X$ is a contact point of escribed circle of $\triangle AA_1A_2$ with side $A_1A_2$. Indeed, consider a homothety with center $A$ which maps incircle $\omega$ of $\triangle AA_1A_2$ to its escribed circle. This homothety maps the line that is tangent to $\omega$ in point $E$ to the parallel line which is tangent to the escribed circle, i.e. to the line $A_1A_2$. Therefore the point $E$ maps to the point $X$, hence $A_1A_2$ is tangent to the escribed circle of $\triangle AA_1A_2$ in the point $X$.

One can similarly prove that $X$ is a tangent point of the line $C_1C_2$ and incircle of $\triangle C_1CC_2$. From the first statement we conclude that $A_1X = FA_2$, and from the second one that $C_1X = FC_2$. It remains to subtract the second equality from the first one.