SHARYGIN 2013 Problem 15

(a) Triangles $A_1B_1C_1$ and $A_2B_2C_2$ are inscribed into triangle $ABC$ so that $C_1A_1 \perp BC$ , $A_1B_1 \perp CA$ , $B_1C_1 \perp AB$ , $B_2A_2 \perp BC$ , $C_2B_2 \perp CA$ , $A_2C_2 \perp AB$ . Prove that these triangles are equal.

(b) Points $A_1$ , $B_1$ , $C_1$ , $A_2$ , $B_2$ , $C_2$ lie inside a triangle $ABC$ so that $A_1$ is on segment $AB_1$ , $B_1$ is on segment $BC_1$ , $C_1$ is on segment $CA_1$ , $A_2$ is on segment $AC_2$ , $B_2$ is on segment $BA_2$ , $C_2$ is on segment $CB_2$ , and the angles $BAA_1$ , $CBB_2$ , $ACC_1$ , $CAA_2$ , $ABB_2$ , $BCC_2$ are equal. Prove that the triangles $A_1B_1C_1$ and $A_2B_2C_2$ are equal.