2017 USAJMO Problems/Problem 3

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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 $ABC$.

[asy]     size(3inch);     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]

Solution

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See also

2017 USAJMO (ProblemsResources)
Preceded by
Problem 2
Followed by
Problem 4
1 2 3 4 5 6
All USAJMO Problems and Solutions
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