USAJMO 2011 Problem 3

For a point $P = (a,a^2)$ in the coordinate plane, let $l(P)$ denote the line passing through $P$ with slope $2a$ . Consider the set of triangles with vertices of the form $P_1 = (a_1, a_1^2), P_2 = (a_2, a_2^2), P_3 = (a_3, a_3^2)$ , such that the intersection of the lines $l(P_1), l(P_2), l(P_3)$ form an equilateral triangle $\triangle$ . Find the locus of the center of $\triangle$ as $P_1P_2P_3$ ranges over all such triangles.