Question 152
In triangle $ABC$ with incentre $I$, let $M_A, M_B$ and $M_C$ be the midpoints of $BC, CA$ and $AB$ respectively, and $H_A, H_B$ and $H_C$ be the feet of altitudes from $A, B$ and $C$ to the respective sides. Denote by $\ell_b$ the line being tangent to the circumcircle of triangle $ABC$ and passing through $B$, and denote by $\ell'_b$ the reflection of $\ell_b$ in $BI$. Let $P_B$ be the intersection of $M_A M_C$ and $\ell_b$, and let $Q_B$ be the intersection of $H_A H_C$ and $\ell'_b$. Define $\ell_c, \ell'_c, P_C$ and $Q_C$ analogously. If $R$ is the intersection of $P_B Q_B$ and $P_C Q_C$, prove that $RB = RC$.