Question 6
Triangle $ABC$ is not isosceles at $A$, and its angles at $B,C$ are acute. A variable point $D$ lies on segment $BC$, distinct from $B,C$ and the foot of the perpendicular from $A$ to $BC$. The line $d$ perpendicular to $BC$ at $D$ meets the lines $AB,AC$ at $E,F$, respectively. Let $M,N,P$ be the incenters of triangles $AEF,BDE,CDF$, respectively. Prove that $A,M,N,P$ are concyclic if and only if $d$ passes through the incenter of triangle $ABC$.