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Difference between revisions of "2017 USAJMO Problems/Problem 3"
(→‎Solution)
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==Solution==
 
==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>
 
{{MAA Notice}}
 
{{MAA Notice}}
  
 
==See also==
 
==See also==
 
{{USAJMO newbox|year=2017|num-b=2|num-a=4}}
 
{{USAJMO newbox|year=2017|num-b=2|num-a=4}}

Revision as of 18:14, 19 April 2017

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. AMC logo.png

See also

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