Question 180
Let $BE$ and $CF$ be altitudes in a tetrahedron $ABCD$. A plane $\alpha$ through the midpoint of $AD$ is perpendicular to $AD$. Assume that the points $A$, $C$, $D$, and $E$ lie on a circle, and the points $A$, $B$, $D$, and $F$ also lie on a circle. Show that the points $E$ and $F$ are equidistant from $\alpha$. (A. Kuznetsov)