VMO 2018 Problem 2

We have a scalene acute triangle $ABC$ (triangle with no two equal sides) and a point $D$ on side $BC$ . Pick a point $E$ on side $AB$ and a point $F$ on side $AC$ such that $\angle DEB=\angle DFC$ . Lines $DF, DE$ intersect $AB, AC$ at points $M, N$ , respectively. Denote $(I_1), (I_2)$ by the circumcircles of triangles $DEM, DFN$ in that order. 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 $H$ . $P$ is the intersection of $(I_1), (I_2)$ different from $D$ . $Q$ is the intersection of $(J_1), (J_2)$ different from $D$ . a. Prove that all points $D, P, Q$ lie on the same line. b. The circumcircles of triangles $AEF, AHK$ intersect at $A, G$ . $(AEF)$ also cut $AQ$ at $A, L$ . Prove that the tangent at $D$ of $(DQG)$ cuts $EF$ at a point on $(DLG)$ .