Question 166
Given an acute, scalene triangle $ABC$, $D$ is a point on side $BC$. Let $E$, $F$ be the points on $AB$, $AC$ such that $\angle DEB = \angle DFC$. Lines $DF$, $DE$ intersect $AB$, $AC$ at points $M$, $N$, respectively. Denote by $(I_1)$, $(I_2)$ the circumcircles of triangles $DEM$, $DFN$. The circle $(J_1)$ touches $(I_1)$ internally at $D$ and touches $AB$ at $K$, circle $(J_2)$ touches $(I_2)$ internally at $D$ and touches $AC$ at $H$. Let $P$ be the intersection of $(I_1)$, $(I_2)$ and $Q$ be the intersection of $(J_1)$, $(J_2)$ ($P, Q \neq D$).
a) Prove that $D$, $P$, and $Q$ are collinear.
b) The circumcircle of triangle $AEF$ meets the circumcircle of triangle $AHK$ again at $G$ and meets the line $AQ$ again at $L$. Prove that the tangent line from $D$ of the circumcircle of triangle $DQG$ intersects $EF$ at a point on the circumcircle of triangle $DLG$.
a) Prove that $D$, $P$, and $Q$ are collinear.
b) The circumcircle of triangle $AEF$ meets the circumcircle of triangle $AHK$ again at $G$ and meets the line $AQ$ again at $L$. Prove that the tangent line from $D$ of the circumcircle of triangle $DQG$ intersects $EF$ at a point on the circumcircle of triangle $DLG$.