Question 200
Point $D$ is chosen on the *Euler line* of triangle $ABC$, inside the triangle. Let $E, F$ be the intersection points of $BD, AC$ and $CD, AB$, respectively. Point $X$ lies on the line $AD$ such that $\angle EXF = 180^\circ - \angle A$, also $A$ and $X$ are on the same side of $EF$. If $P$ is the second intersection of circumcircles of $CXF$ and $BXE$, prove that the lines $XP$ and $EF$ meet on the altitude of $A$. (the Euler line is the line connecting the orthocenter to the circumcenter.)
