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Locus of points P in the plane of an equilateral triangle

Source: 1976 AHSME Problem 22

May 19, 2014
analytic geometryAMC

Problem Statement

Given an equilateral triangle with side of length ss, consider the locus of all points P\mathit{P} in the plane of the triangle such that the sum of the squares of the distances from P\mathit{P} to the vertices of the triangle is a fixed number aa. This locus
<spanclass=′latex−bold′>(A)</span>is a circle if a>s2<span class='latex-bold'>(A) </span>\text{is a circle if }a>s^2\qquad
<spanclass=′latex−bold′>(B)</span>contains only three points if a=2s2 and is a circle if a>2s2<span class='latex-bold'>(B) </span>\text{contains only three points if }a=2s^2\text{ and is a circle if }a>2s^2\qquad
<spanclass=′latex−bold′>(C)</span>is a circle with positive radius only if s2<a<2s2<span class='latex-bold'>(C) </span>\text{is a circle with positive radius only if }s^2<a<2s^2\qquad
<spanclass=′latex−bold′>(D)</span>contains only a finite number of points for any value of a<span class='latex-bold'>(D) </span>\text{contains only a finite number of points for any value of }a\qquad
<spanclass=′latex−bold′>(E)</span>is none of these<span class='latex-bold'>(E) </span>\text{is none of these}
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