Question 4
Let $ABC$ be an acute, non-isosceles triangle with altitude $AD$, and let $P$ lie on $AD$. Lines $PB,PC$ meet $CA,AB$ at $E,F$, respectively.
(a) If $AEDF$ is cyclic, prove that $\dfrac{PA}{PD}=(\tan B+\tan C)\cot\dfrac A2$.
(b) Let $M$ be the intersection of $CP$ with the line through $B$ perpendicular to $AB$, and $N$ the intersection of $BP$ with the line through $C$ perpendicular to $AC$. Let $K$ be the perpendicular projection of $A$ onto $MN$. Prove that $\angle BKC+\angle MAN$ is constant as $P$ moves along $AD$.
All intersections and angles are understood in nondegenerate positions. Source-domain caveat: part (b), with ordinary angles on the whole altitude segment, is not valid as printed; the solution preserves the source’s configuration argument and gives an exact countercheck.
(a) If $AEDF$ is cyclic, prove that $\dfrac{PA}{PD}=(\tan B+\tan C)\cot\dfrac A2$.
(b) Let $M$ be the intersection of $CP$ with the line through $B$ perpendicular to $AB$, and $N$ the intersection of $BP$ with the line through $C$ perpendicular to $AC$. Let $K$ be the perpendicular projection of $A$ onto $MN$. Prove that $\angle BKC+\angle MAN$ is constant as $P$ moves along $AD$.
All intersections and angles are understood in nondegenerate positions. Source-domain caveat: part (b), with ordinary angles on the whole altitude segment, is not valid as printed; the solution preserves the source’s configuration argument and gives an exact countercheck.