Question 3
Triangle $ABC$ has incircle $(I)$ tangent to $BC,CA,AB$ at $D,E,F$, respectively. Let $I_b,I_c$ be its $B$- and $C$-excenters. Let $P,Q$ be the midpoints of $I_bE,I_cF$. The circles $(PAC),(QAB)$ meet $AB,AC$ again at $R,S$, respectively.
(a) Prove that $PR,QS,AI$ are concurrent.
(b) Put $K=DE\cap I_bI_c$, $J=DF\cap I_bI_c$, and $M=EJ\cap FK$. The lines $PE,QF$ meet $(PAC),(QAB)$ again at $X,Y$, respectively. Prove that $BY,CX,AM$ are concurrent.
(a) Prove that $PR,QS,AI$ are concurrent.
(b) Put $K=DE\cap I_bI_c$, $J=DF\cap I_bI_c$, and $M=EJ\cap FK$. The lines $PE,QF$ meet $(PAC),(QAB)$ again at $X,Y$, respectively. Prove that $BY,CX,AM$ are concurrent.