The Problems of Ukrainian Authors

On the sides $AB$, $BC$, $CA$ of triangle $ABC$ such points $C_1$, $A_1$, $B_1$ are chosen that the lines $A{A}_1$, $B{B}_1$ and $C{C}_1$ meet at point $K$. The perpendiculars are dropped from the point $K$ onto the sides of triangle. Lines $l_1$, $l_2$, $l_3$ are drawn through the feet of these perpendiculars, parallel to lines, symmetrical to $A{A}_1$, $B{B}_1$ and $C{C}_1$ with respect to angular bisectors of $\angle A$, $\angle B$ and $\angle C$ respectively. Prove that the lines $l_1$, $l_2$, $l_3$ meet at a point.

Fig.28