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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.

Monge's plan

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 !

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You've tried taking the coordinate geometry route? – J. M. Oct 26 '11 at 15:02
I tried but there are too many unknowns and I don't see how to characterize this kind of conditions. – Skydreamer Oct 26 '11 at 15:11
The strategy I had in mind was for you to take three of those planes, find their intersection, and then verify that that point lies in those three other planes... – J. M. Oct 26 '11 at 15:15
Okay, this might help you with simplifying things: position your tetrahedron such that one vertex is at the origin and one of the edges lies on an axis. Have one of the faces lie in a coordinate plane, if needed. You might want to use the formulae here to help you with assembling plane equations. – J. M. Oct 26 '11 at 15:29
up vote 2 down vote accepted

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

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I'm watching this tonight but your second link seems dead... – Skydreamer Oct 26 '11 at 18:16
Sorry, I grabbed a redirected link that contained a session code. Fixed. By the way, you're looking at this tonight. "Watching" is for observing something that may change over time, like a process or a movie or a game or a sunset. – joriki Oct 26 '11 at 18:22
Oh, I didn't know :) I'm not english ! Thank you. – Skydreamer Oct 26 '11 at 18:59

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