USAJMO 2011 Question 3

For $P=(a,a^2)$ let $\ell(P)$ be the line through $P$ with slope $2a$. Consider triples of distinct points $P_i=(a_i,a_i^2)$ whose three lines $\ell(P_i)$ form an equilateral triangle $\Delta$. Find the locus of the center of $\Delta$.

Proposed by Zuming Feng (credit in the supplied notes).