2001 Iran NMO (Round 3) Problems/Problem 5
Problem
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.
Solution
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.
Resources
- 2001 Iran NMO (Round 3)
- <url>Forum/viewtopic.php?t=6528 Discussion on AoPS/MathLinks</url>