Baltic Way shortlist

Denote by $X$ the intersection point of the lines $A_1{A}_2$ and $AC$. Prove that $X$ is a contact point of escribed circle of $△A{A}_1{A}_2$ with side $A_1{A}_2$. Indeed, consider a homothety with center $A$ which maps incircle $\omega$ of $△A{A}_1{A}_2$ 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 $A_1{A}_2$. Therefore the point $E$ maps to the point $X$, hence $A_1{A}_2$ is tangent to the escribed circle of $△A{A}_1{A}_2$ in the point $X$.



One can similarly prove that $X$ is a tangent point of the line $C_1{C}_2$ and incircle of $△C_{1}C{C}_2$. From the first statement we conclude that $A_{1}X=F{A}_2$, and from the second one that $C_{1}X=F{C}_2$. It remains to subtract the second equality from the first one.