Question 23
Let $P$ be a point on the side $BC$ of an acute triangle $ABC$. Let $H$ be the orthocenter of $\triangle ABC$ and $D$ be the foot of the perpendicular from $H$ to $AP$. Let $\Gamma_1, \Gamma_2$ be the circumcircles of $\triangle ABD, \triangle ACD$, respectively. Let $l$ be the line parallel to $BC$ and passing through $D$. $l$ intersects $\Gamma_1$ and $\Gamma_2$ at $X$ and $Y$, different from $D$, respectively. $l$ intersects $AB$ and $AC$ at $E$ and $F$, respectively. Two lines $XB$ and $YC$ meet at $Z$. Show that $BP = CP$ if and only if $ZE = ZF$.
