Let $\Gamma_B$ and $\Gamma_C$ be excircles of an acute-angled triangle $ABC$ opposite to its vertices $B$ and $C$, respectively. Let $C_1$ and $L$ be the tangent points of $\Gamma_C$ and the side $AB$ and the line $BC$ respectively. Let $B_1$ and $M$ be the tangent points of $\Gamma_B$ and the side $AC$ and the line $BC$, respectively. Let $X$ be the point of intersection of the lines $LC_1$ and $MB_1$. Prove that $AX$ is equal to the inradius of the triangle $ABC$. (A. Voidelevich)