SHARYGIN 2011 Problem 23

Given are triangle $ABC$ and line $\ell$ intersecting $BC, CA$ and $AB$ at points $A_1, B_1$ and $C_1$ respectively. Point $A'$ is the midpoint of the segment between the projections of $A_1$ to $AB$ and $AC$ . Points $B'$ and $C'$ are defined similarly. (a) Prove that $A', B'$ and $C'$ lie on some line $\ell'$ . (b) Suppose $\ell$ passes through the circumcenter of $\triangle ABC$ . Prove that in this case $\ell'$ passes through the center of its nine-points circle.

M. Marinov and N. Beluhov