Question 2
Let $ABC$ be an acute scalene triangle and $D$ a point on side $BC$. Choose $E$ on side $AB$ and $F$ on side $AC$ so that $\angle DEB=\angle DFC$. The lines $DF,DE$ meet $AB,AC$ at $M,N$, respectively. Let $(I_1),(I_2)$ be the circumcircles of $DEM,DFN$, respectively. Let $(J_1)$ be internally tangent to $(I_1)$ at $D$ and tangent to $AB$ at $K$, and let $(J_2)$ be internally tangent to $(I_2)$ at $D$ and tangent to $AC$ at $H$. Let $P\ne D$ be the other intersection of $(I_1),(I_2)$, and $Q\ne D$ the other intersection of $(J_1),(J_2)$.
(a) Prove that $D,P,Q$ are collinear.
(b) The circumcircle of $AEF$ meets the circumcircle of $AHK$ and the line $AQ$ again at $G,L$, respectively, with $G,L\ne A$. Prove that the tangent at $D$ to the circumcircle of $DQG$ meets $EF$ at a point on the circumcircle of $DLG$.
(a) Prove that $D,P,Q$ are collinear.
(b) The circumcircle of $AEF$ meets the circumcircle of $AHK$ and the line $AQ$ again at $G,L$, respectively, with $G,L\ne A$. Prove that the tangent at $D$ to the circumcircle of $DQG$ meets $EF$ at a point on the circumcircle of $DLG$.