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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
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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
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1 Answer

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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