2014 USAJMO Problems/Problem 2
Problem
Let 
be a non-equilateral, acute triangle with $\angle A=60\textdegrees$ (Error compiling LaTeX. Unknown error_msg), and let 
and 
denote the circumcenter and orthocenter of , respectively.
(a) Prove that line 
intersects both segments 
and 
.
(b) Line intersects segments and at 
and 
, respectively. Denote by 
and 
the respective areas of triangle 
and quadrilateral 
. Determine the range of possible values for 
.
Solution
![[asy] import olympiad; unitsize(1inch); pair A,B,C,O,H,P,Q,i1,i2,i3,i4; //define dots A=3*dir(50); B=(0,0); C=right*2.76481496; O=circumcenter(A,B,C); H=orthocenter(A,B,C); i1=2*O-H; i2=2*i1-O; i3=2*H-O; i4=2*i3-H; //These points are for extending PQ. DO NOT DELETE! P=intersectionpoint(i2--i4,A--B); Q=intersectionpoint(i2--i4,A--C); //draw dot(P); dot(Q); draw(P--Q); dot(A); dot(B); dot(C); dot(O); dot(H); draw(A--B--C--cycle); //label label("$A$",A,N); label("$B$",B,SW); label("$C$",C,SE); label("$P$",P,NW); label("$Q$",Q,NE); label("$O$",O,N); label("$H$",H,N); //change O and H label positions if interfering with other lines to be added //further editing: ABCPQOH are the dots to be further used. i1,i2,i3,i4 are for drawing assistence and are not to be used [/asy]](http://latex.artofproblemsolving.com/9/7/7/977c60b02738b163d848b5c00b0e86c290c2db98.png)
