1973 IMO Shortlist Problems/Bulgaria 1

Problem

A tetrahedron $\displaystyle ABCD$ is inscribed in the sphere $\displaystyle S$ . Find the locus of points $\displaystyle P$ , situated in $\displaystyle S$ , such that

Let $\displaystyle S$ have center $\displaystyle O = (0,0,0)$ and radius $\displaystyle r$ . Since the power of $\displaystyle P$ with respect to $\displaystyle S$ is invariant, we may multiply both sides of the condition by that power to obtain

We may now use the law of cosines to rewrite the condition thus:

If we now complete the square for $\displaystyle x, y$ , and $\displaystyle z$ , it becomes apparent that this is an equation for a sphere centered at the midpoint of the segment with endpoints $\displaystyle O$ and the centroid of $\displaystyle ABCD$ and with radius half the distance from $\displaystyle O$ and the centroid of $\displaystyle ABCD$ , which is therefore the desired locus, Q.E.D.

Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.