2017 USAJMO Problems/Problem 3: Difference between revisions
No edit summary |
|||
| Line 1: | Line 1: | ||
==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>. | ||
<asy> | <asy> | ||
size( | 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); | ||
| Line 21: | Line 19: | ||
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 18:17, 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 .
![[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]](http://latex.artofproblemsolving.com/7/8/c/78c3af92031ce82e11f57a21c37bc147a0b11852.png)
Solution
These problems are copyrighted © by the Mathematical Association of America. 
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 | ||
