3rd VMO 1964 Problem A3 Let O be a point not in the plane p and A a point in p. For each line in p through A, let H be the foot of the perpendicular from O to the line. Find the locus of H.
circle diameter AB, where OB is the normal to p Solution Let B be the foot of the perpendicular from O to p. We claim that the locus is the circle diameter AB. Any line in p through A meets this circle at one other point K (except for the tangent to the circle at A, but in that case A is obviously the foot of the perpendicular from O to the line). Now BK is perpendicular to AK, so OK is also perpendicular to AK, and hence K must be the foot of the perpendicular from O to the line. 3rd VMO 1964 © John Scholes jscholes@kalva.demon.co.uk 7 March 2004 Last corrected/updated 7 Mar 04