VN-TST 2018 Problem 6

Triangle $ABC$ circumscribed $(O)$ has $A$ -excircle $(J)$ that touches $AB,\ BC,\ AC$ at $F,\ D,\ E$ , resp.

a. $L$ is the midpoint of $BC$ . Circle with diameter $LJ$ cuts $DE,\ DF$ at $K,\ H$ . Prove that $(BDK),\ (CDH)$ has an intersecting point on $(J)$ . b. Let $EF\cap BC =\{G\}$ and $GJ$ cuts $AB,\ AC$ at $M,\ N$ , resp. $P\in JB$ and $Q\in JC$ such that $$\angle PAB=\angle QAC=90{}^\circ .$$ $PM\cap QN=\{T\}$ and $S$ is the midpoint of the larger $BC$ -arc of $(O)$ . $(I)$ is the incircle of $ABC$ . Prove that $SI\cap AT\in (O)$ .