Question 122
Let $ABC$ be a triangle and let $A_{1}$, $B_{1}$ and $C_{1}$ be the points of tangency of the $A$-excircle, denoted $\omega_{A}$, with the side $BC$ and the rays $[AC)$ and $[AB)$ respectively. Let $P$ be the midpoint of the segment $[B_{1}C_{1}]$. The line $(A_{1}P)$ meets the circle $\omega_{A}$ again at the point $X$. The tangent to the circumcircle of triangle $ABC$ at the point $A$ and the tangent to the circle $\omega_{A}$ at the point $X$ meet at $R$. Show that $RX = RP$.