Question 23
A triangle $ABC$ is not isosceles at $A$, and its angles $\angle ABC$, $\angle ACB$ are acute. Consider a point $D$ moving on edge $BC$, so that $D$ does not coincide with $B$, $C$ and with the perpendicular projection of $A$ on $BC$. The line $d$, perpendicular with $BC$ at $D$, intersects the lines $AB$ and $AC$ at $E$ and $F$, respectively. Let $M$, $N$, and $P$ be the incenters of the triangles $AEF$, $BDE$ and $CDF$, respectively. Show that four points $A, M, N, P$ lie on one circle if and only if the line $d$ passes through the incenter of triangle $ABC$.