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Plane passing through vertex of regular tetrahedron

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November 3, 2010
geometry3D geometrytetrahedrongeometry unsolved

Problem Statement

(a)(a) A plane π\pi passes through the vertex OO of the regular tetrahedron OPQROPQR. We define p,q,rp, q, r to be the signed distances of P,Q,RP,Q,R from π\pi measured along a directed normal to π\pi. Prove that p2+q2+r2+(q−r)2+(r−p)2+(p−q)2=2a2,p^2 + q^2 + r^2 + (q - r)^2 + (r - p)^2 + (p - q)^2 = 2a^2, where aa is the length of an edge of a tetrahedron. (b)(b) Given four parallel planes not all of which are coincident, show that a regular tetrahedron exists with a vertex on each plane.
Note: Part (b)(b) is [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=49&t=60825&start=0]IMO 1972 Problem 6
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