2005 USAMO Problems/Problem 3
(Zuming Feng) Let be an acute-angled triangle, and let and be two points on side . Construct point in such a way that convex quadrilateral is cyclic, , and and lie on opposite sides of line . Construct point in such a way that convex quadrilateral is cyclic, , and and lie on opposite sides of line . Prove that points , and lie on a circle.
Let be the second intersection of the line with the circumcircle of , and let be the second intersection of the circumcircle of and line . It is enough to show that and . All our angles will be directed, and measured mod .
Since points are concyclic and points are collinear, it follows that But since points are concyclic, It follows that lines and are parallel. If we exchange with and with in this argument, we see that lines and are likewise parallel.
It follows that is the intersection of and the line parallel to and passing through . Hence . Then is the second intersection of the circumcircle of and the line parallel to passing through . Hence , as desired.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
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