Collinearity problem (Newton-Gauss line)

Question

I had some troubles with this problem :

Let $ABCD$ be a convex quadrilateral. $M$ and $N$ are the midpoints of the diagonals $AC$ and $BD$. The sides $AB$ and $CD$ are extended until they intersect. The intersection point is $E$. The sides $AD$ and $BC$ are extended until they intersect. The intersection point is $F$. Let $P$ be the midpoint of the segment $[EF]$. Prove that $M$, $N$, $P$ are collinear.

First, I found that a quadrilateral in which the opposite sides interesect is also known as a complete quadrilateral. Then, the line $M-N-P$ is known as Newton-Gauss line and the problem above as Newton's Problem.

I've taught about solving it using areas. I've used the property that median divides the triangle in two echivalent triangles (with the same area). Many properties can be derived from it.

I haven't figure out, but I'm interested in a proof using areas. I would appreciate some suggestions.

Thanks!

Answer 1

In fact there is an alternate, nice, proof using areas that I got, using a link found in the Wikipedia article, in the book by Johnson, "Modern geometry; An Elementary Treatise on the Geometry of the Triangle and the Circle, Houghton Mifflin Ed., 1929" available online (public domain), page 61-62. I reproduce here the two pages.

Nice, isn't it ?