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2011 USAJMO Problems/Problem 3

Revision as of 16:17, 30 April 2011 by Remy1140 (talk | contribs) (Created page with '== Problem == For a point <math>P = (a, a^2)</math> in the coordinate plane, let <math>\ell(P)</math> denote the line passing through <math>P</math> with slope <math>2a</math>. …')
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Problem

For a point $P = (a, a^2)$ in the coordinate plane, let $\ell(P)$ 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 intersections of the lines $\ell(P_1)$, $\ell(P_2)$, $\ell(P_3)$ form an equilateral triangle $\Delta$. Find the locus of the center of $\Delta$ as $P_1P_2P_3$ ranges over all such triangles.

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