Question 164
Let $ABC$ be a triangle with $I$ as its incenter and the circle $(I)$ is tangent to $BC$, $CA$, $AB$ at $D$, $E$, $F$ respectively. Denote $I_b$, $I_c$ as the excenters of triangle $ABC$ with respect to vertices $B$, $C$. Let $P$, $Q$ be the midpoints of segments $I_bE$, $I_cF$. Suppose that $(PAC)$ intersects $AB$ at the second point $R$ and $(QAB)$ intersects $AC$ at the second point $S$.
1. Prove that $PR$, $QS$, $AI$ are concurrent.
2. Suppose that $DE$, $DF$ intersect $I_bI_c$ at $K$, $J$ and $EJ$ meets $FK$ at $M$. The lines $PE$, $QF$ intersect $(PAC)$, $(QAB)$ at $X$, $Y$ ($X$ differs from $P$ and $Y$ differs from $Q$). Prove that $BY$, $CX$ and $AM$ are concurrent.
1. Prove that $PR$, $QS$, $AI$ are concurrent.
2. Suppose that $DE$, $DF$ intersect $I_bI_c$ at $K$, $J$ and $EJ$ meets $FK$ at $M$. The lines $PE$, $QF$ intersect $(PAC)$, $(QAB)$ at $X$, $Y$ ($X$ differs from $P$ and $Y$ differs from $Q$). Prove that $BY$, $CX$ and $AM$ are concurrent.