Given a point A and a segment BC, determine the locus of all points P in space for which ∠APX = 90o for some X on the segment BC.
Solution
Take the solid sphere on diameter AB, and the solid sphere on diameter AC. Then the locus is the points in one sphere but not the other (or on the surface of either sphere). Given P, consider the plane through P perpendicular to AP and the parallel planes through the other two points of intersection of AP with the two spheres (apart from A) which pass through B and C.
Solutions are also available in: Samuel L Greitzer, International Mathematical Olympiads 1959-1977, MAA 1978, and in István Reiman, International Mathematical Olympiad 1959-1999, ISBN 189-8855-48-X.
© John Scholes
jscholes@kalva.demon.co.uk
22 Sep 1998
Last corrected/updated 24 Sep 2003