How to determine the intersection of 6 planes?

Question

ABCD is a tetrahedron (not necessarly a regular one). A Monge's plane is a plane which is perpendicular to an edge and goes through the middle of the opposite edge.

I want to prove that the 6 Monge's planes of this triangle converge in a unique point and I haven't got any idea of the way to do this.

Thank you in advance !

Answer 1

Here's a concise linear algebra proof (pages 5 and 6). And here's a geometrical proof.