2017 USAJMO Problems/Problem 3

Problem

( $*$ ) Let $ABC$ be an equilateral triangle and let $P$ be a point on its circumcircle. Let lines $PA$ and $PB$ intersect at $D$ ; let lines $PB$ and $CA$ intersect at $E$ ; and let lines $PC$ and $AB$ intersect at $F$ . Prove that the area of triangle $DEF$ is twice that of triangle $ABC$ .

Solution

$[asy] size(5inch); pair A = (0, 3sqrt(3)), B = (-3,0), C = (3,0), P = (0, -sqrt(3)), D = (0, 0), E1 = (6, -3sqrt(3)), F = (-6, -3sqrt(3)), O = (0, sqrt(3)); draw(Circle(O, 2sqrt(3)), black); draw(A--B--C--cycle); draw(B--E1--C); draw(C--F--B); draw(A--P); draw(D--E1--F--cycle, dashed); label("A", A, N); label("B", B, W); label("C", C, E); label("P", P, S); label("D", D, NW); label("E", E1, SE); label("F", F, SW); [/asy]$ The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

2017 USAJMO (Problems • Resources)
Preceded by Problem 2	Followed by Problem 4
1 • 2 • 3 • 4 • 5 • 6
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2017 USAJMO Problems/Problem 3

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