SHARYGIN 2015 Problem 24

The insphere of a tetrahedron ABCD with center $O$ touches its faces at points $A_1,B_1,C_1$ and $D_1$ . a) Let $P_a$ be a point such that its reflections in lines $OB,OC$ and $OD$ lie on plane $BCD$ . Points $P_b, P_c$ and $P_d$ are defined similarly. Prove that lines $A_1P_a,B_1P_b,C_1P_c$ and $D_1P_d$ concur at some point $P$ . b) Let $I$ be the incenter of $A_1B_1C_1D_1$ and $A_2$ the common point of line $A_1I$ with plane $B_1C_1D_1$ . Points $B_2, C_2, D_2$ are defined similarly. Prove that $P$ lies inside $A_2B_2C_2D_2$ .