1973 IMO Shortlist Problems/Bulgaria 1
A tetrahedron is inscribed in the sphere . Find the locus of points , situated in , such that
where are the other intersection points of with .
Let have center and radius . Since the power of with respect to 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 let , then we may rewrite the expression (using dot products) thus:
If we now complete the square for , and , it becomes apparent that this is an equation for a sphere centered at the midpoint of the segment with endpoints and the centroid of and with radius half the distance from and the centroid of , which is therefore the desired locus, Q.E.D.
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