a geometry problem Let $DA$ be the normal on the plane of the triangle $ABC$ and $E \in (DA)$. Let's notate with $M,N,P,Q$ the proiections of the point $A$ to the lines $BD$, $CD$, $BE$, respectively $CE$. 
Prove that: 

1) $$MN\cap BC \cap PQ \neq \emptyset;$$
  2) the quadrilateral $MNPQ$ is inscriptible.  


For 2), consider the circumscribed circle of the $ABC$ triangle, and expand it to a sphere in 3d with the same center and radius. If we cut it by a plane orthogonal to the $ABC$ plane and containing $AB$, we get a circle with $AB$ as its diameter, so, by Thales' thm the points $M$ and $P$ will be on that circle, hence on the sphere. Similarly for $N,Q$. Well, it's still needed that these 4 points are in the same plane..
So, for the rest, I would use vectors (or coordinate geometry), setting up the coordinate system in a preferable way, say $A$ is the origo, $ABC$ plane is the $x,y$-plane, we can set also $AB=(1,0,0)$ if it helps..
