2001 Iran NMO (Round 3) Problems/Problem 5
In triangle , let be the incenter and the excenter opposite . Suppose that meets and the circumcircle of triangle at and , respectively. Let be the midpoint of arc of the circumcircle of triangle . Let lines and intersect the circumcircle of triangle again at and , respectively. Prove that , , and are collinear.
We will use directed angles mod , and directed arcs mod .
Since , it follows that It follows that quadrilateral is cyclic.
On the other hand, , so quadrilateral is cyclic.
Now, since is the radical axis of the circumcircles of and , is the radical axis of the circumcircles of and , and is the circumcircle of and , these three lines concur at , as desired.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
- 2001 Iran NMO (Round 3)
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