Difference between revisions of "2017 USAJMO Problems/Problem 3"

(Solution)
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==Problem==
 
==Problem==
 
(<math>*</math>) Let <math>ABC</math> be an equilateral triangle and let <math>P</math> be a point on its circumcircle. Let lines <math>PA</math> and <math>PB</math> intersect at <math>D</math>; let lines <math>PB</math> and <math>CA</math> intersect at <math>E</math>; and let lines <math>PC</math> and <math>AB</math> intersect at <math>F</math>. Prove that the area of triangle <math>DEF</math> is twice that of triangle <math>ABC</math>.
 
(<math>*</math>) Let <math>ABC</math> be an equilateral triangle and let <math>P</math> be a point on its circumcircle. Let lines <math>PA</math> and <math>PB</math> intersect at <math>D</math>; let lines <math>PB</math> and <math>CA</math> intersect at <math>E</math>; and let lines <math>PC</math> and <math>AB</math> intersect at <math>F</math>. Prove that the area of triangle <math>DEF</math> is twice that of triangle <math>ABC</math>.
 
==Solution==
 
  
 
<asy>
 
<asy>
     size(5inch);
+
     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));
 
     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(Circle(O, 2sqrt(3)), black);
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     label("F", F, SW);
 
     label("F", F, SW);
 
</asy>
 
</asy>
 +
 +
==Solution==
 +
 
{{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 19:17, 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 $ABC$.

[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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

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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