RU-ORAL-GEOMETRY 2013

In triangle $ABC$ the angle bisector $AK$ is perpendicular on the median is $CL$ . Prove that in the triangle $BKL$ also one of angle bisectors are perpendicular to one of the medians.

With a compass and a ruler, split a triangle into two smaller triangles with the same sum of squares of sides.

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 $AC$ .