RU-ORAL-GEOMETRY 2013 Problem 3

The bisectors $AA_1$ and $CC_1$ of the right triangle $ABC$ ( $\angle B = 90^o$ ) intersect at point $I$ . The line passing through the point $C_1$ and perpendicular on the line $AA_1$ intersects the line that passes through $A_1$ and is perpendicular on $CC_1$ , at the point $K$ . Prove that the midpoint of the segment $KI$ lies on segment $AC$ .