Difference between revisions of "2017 USAJMO Problems/Problem 3"
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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 
be an equilateral triangle and let 
be a point on its circumcircle. Let lines 
and 
intersect at 
; let lines and 
intersect at 
; and let lines 
and 
intersect at 
. Prove that the area of triangle 
is twice that of triangle .
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]](http://latex.artofproblemsolving.com/f/8/3/f83750c2abd7e3597a04b52eaa9422d0a1ad471c.png)
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
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 | ||
