SHARYGIN 2011 Problem 16

Given are triangle $ABC$ and line $\ell$ . The reflections of $\ell$ in $AB$ and $AC$ meet at point $A_1$ . Points $B_1, C_1$ are defined similarly. Prove that a) lines $AA_1, BB_1, CC_1$ concur, b) their common point lies on the circumcircle of $ABC$ c) two points constructed in this way for two perpendicular lines are opposite.