Question 20
Let $A$, $B$, $C$ be the midpoints of the sides $B'C'$, $C'A'$, $A'B'$ of $\triangle A'B'C'$, respectively. Let the point $P$ lie inside $\triangle ABC$, and let $AP$, $BP$, $CP$ meet $BC$, $CA$, $AB$ at the points $P_a$, $P_b$, $P_c$, respectively.
The lines $P_aP_b$, $P_aP_c$ meet $B'C'$ at $R_b$, $R_c$, respectively; the lines $P_bP_c$, $P_bP_a$ meet $C'A'$ at $S_c$, $S_a$, respectively; the lines $P_cP_a$, $P_cP_b$ meet $A'B'$ at $T_a$, $T_b$, respectively. It is known that $S_c$, $S_a$, $T_a$, $T_b$ all lie on a circle with centre $O$.
Prove that $OR_b = OR_c$.
The lines $P_aP_b$, $P_aP_c$ meet $B'C'$ at $R_b$, $R_c$, respectively; the lines $P_bP_c$, $P_bP_a$ meet $C'A'$ at $S_c$, $S_a$, respectively; the lines $P_cP_a$, $P_cP_b$ meet $A'B'$ at $T_a$, $T_b$, respectively. It is known that $S_c$, $S_a$, $T_a$, $T_b$ all lie on a circle with centre $O$.
Prove that $OR_b = OR_c$.