Question 78
Let $ABC$ be an acute nonisosceles triangle with incenter $I$ and $(d)$ is an arbitrary line tangent to $(I)$ at $K$. The lines passes through $I$, perpendicular to $IA$, $IB$, $IC$ cut $(d)$ at $A_{1}$, $B_{1}$, $C_{1}$ respectively. Suppose that $(d)$ cuts $BC$, $CA$, $AB$ at $M$, $N$, $P$ respectively. The lines through $M$, $N$, $P$ and respectively parallel to the internal bisectors of $A$, $B$, $C$ in triangle $ABC$ meet each other to define a triangle $XYZ$. Prove that three lines $AA_{1}$, $BB_{1}$, $CC_{1}$ are concurrent and $IK$ is tangent to the circle $(XYZ)$.