Question 7
(GBR 1) IMO 6.
(a) A plane $\pi$ passes through the vertex $O$ of the regular tetrahedron $OPQR$. We define $p,q,r$ to be the signed distances of $P,Q,R$ from $\pi$ measured along a directed normal to $\pi$. Prove that
$$p^2+q^2+r^2+(q-r)^2+(r-p)^2+(p-q)^2=2a^2,$$
where $a$ is the length of an edge of the tetrahedron.
(b) Given four parallel planes not all of which are coincident, show that a regular tetrahedron exists with a vertex on each plane.
(a) A plane $\pi$ passes through the vertex $O$ of the regular tetrahedron $OPQR$. We define $p,q,r$ to be the signed distances of $P,Q,R$ from $\pi$ measured along a directed normal to $\pi$. Prove that
$$p^2+q^2+r^2+(q-r)^2+(r-p)^2+(p-q)^2=2a^2,$$
where $a$ is the length of an edge of the tetrahedron.
(b) Given four parallel planes not all of which are coincident, show that a regular tetrahedron exists with a vertex on each plane.