IMO Shortlist 2004 problem G7
Dodao/la:
arhiva2. travnja 2012. For a given triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
, let
![X](/media/m/9/2/8/92802f174fc4967315c2d8002c426164.png)
be a variable point on the line
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
such that
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
lies between
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
and
![X](/media/m/9/2/8/92802f174fc4967315c2d8002c426164.png)
and the incircles of the triangles
![ABX](/media/m/3/d/4/3d4b5eb7119f4d3be1a82da45570a1a7.png)
and
![ACX](/media/m/d/c/7/dc73dae78998dabf0ee217d54afbbbbc.png)
intersect at two distinct points
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
and
![Q.](/media/m/7/8/b/78b8b2b44099003e844b852726990bc5.png)
Prove that the line
![PQ](/media/m/f/2/f/f2f65ec376294df7eca22d2c1a189747.png)
passes through a point independent of
![X](/media/m/9/2/8/92802f174fc4967315c2d8002c426164.png)
.
comment
An extension by Darij Grinberg can be found here.
%V0
For a given triangle $ABC$, let $X$ be a variable point on the line $BC$ such that $C$ lies between $B$ and $X$ and the incircles of the triangles $ABX$ and $ACX$ intersect at two distinct points $P$ and $Q.$ Prove that the line $PQ$ passes through a point independent of $X$.
comment
An extension by Darij Grinberg can be found here.
Izvor: Međunarodna matematička olimpijada, shortlist 2004