59th Ukrainian National Mathematical Olympiad

Circle $k$ of radius $r$ is inscribed in $△ABC$. Tangent lines of $k$, that are parallel to sides $AB$, $BC$ and $CA$, intersect other sides of $△ABC$ at points $M,N$; $P,Q$ and $L,T$ ($P,T\in AB$, $L,N\in BC$ and $M,Q\in AC$). Denote by $r_1,r_2,r_3$ radii of circles inscribed in triangles $MNC,PQA$ and $LTB$, respectively. Prove that $r_1+r_2+r_3=r$.

Fig. 28