1976 AHSME Problem 22

Given an equilateral triangle with side of length s, consider the locus of all points \mathit{P} in the plane of the triangle such that the sum of the squares of the distances from \mathit{P} to the vertices of the triangle is a fixed number a. This locus

\begin{array}{l} \textbf{(A) }\text{is a circle if }a>s^2\qquad\\ \textbf{(B) }\text{contains only three points if }a=2s^2\text{ and is a circle if }a>2s^2\qquad\\ \textbf{(C) }\text{is a circle with positive radius only if }s^2<a<2s^2\qquad\\ \textbf{(D) }\text{contains only a finite number of points for any value of }a\qquad\\ \textbf{(E) }\text{is none of these} \end{array}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.