Question 4
Let $ABCD$ be a cyclic quadrilateral with $DB=DC$. Let $M,N$ be the midpoints of $AB,AC$, and let $J,E,F$ be the contact points of the incircle $(I)$ of triangle $ABC$ with $BC,CA,AB$, respectively. Line $MN$ meets $JE,JF$ at $K,H$, respectively. Line $IJ$ meets circle $(IBC)$ again at $G$, and $DG$ meets $(IBC)$ again at $T$.
a) Prove that $JA$ passes through the midpoint of $HK$ and is perpendicular to $IT$.
b) Let $R,S$ be the perpendicular projections of $D$ on $AB,AC$. Choose $P\in IF$, $Q\in IE$ so that $KP,HQ$ are perpendicular to $MN$. Prove that $MP,NQ,RS$ are concurrent.
a) Prove that $JA$ passes through the midpoint of $HK$ and is perpendicular to $IT$.
b) Let $R,S$ be the perpendicular projections of $D$ on $AB,AC$. Choose $P\in IF$, $Q\in IE$ so that $KP,HQ$ are perpendicular to $MN$. Prove that $MP,NQ,RS$ are concurrent.