VN-TST 2014 Problem 4

a. Let $ABC$ be a triangle with altitude $AD$ and $P$ a variable point on $AD$ . Lines $PB$ and $AC$ intersect each other at $E$ , lines $PC$ and $AB$ intersect each other at $F.$ Suppose $AEDF$ is a quadrilateral inscribed . Prove that $$\frac{PA}{PD}=(\tan B+\tan C)\cot \frac{A}{2}.$$ b. Let $ABC$ be a triangle with orthocentre $H$ and $P$ a variable point on $AH$ . The line through $C$ perpendicular to $AC$ meets $BP$ at $M$ , The line through $B$ perpendicular to $AB$ meets $CP$ at $N.$ $K$ is the projection of $A$ on $MN$ . Prove that $\angle BKC+\angle MAN$ is invariant .