IMO Shortlist 2005 problem G3
Dodao/la:
arhiva2. travnja 2012. Let
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
be a parallelogram. A variable line
![g](/media/m/9/5/8/958b2ae8c90cadb8c953ce50efb9c02a.png)
through the vertex
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
intersects the rays
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
and
![DC](/media/m/3/4/d/34d909d24fdb3a0dd783c13a369556ce.png)
at the points
![X](/media/m/9/2/8/92802f174fc4967315c2d8002c426164.png)
and
![Y](/media/m/3/b/c/3bc24c5af9ce86a9a691643555fc3fd6.png)
, respectively. Let
![K](/media/m/e/1/e/e1ed1943d69f4d6a840e99c7bd199930.png)
and
![L](/media/m/f/c/1/fc1ae4eb78da7d1352cbf1f8217ab286.png)
be the
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
-excenters of the triangles
![ABX](/media/m/3/d/4/3d4b5eb7119f4d3be1a82da45570a1a7.png)
and
![ADY](/media/m/a/0/d/a0df1e21a2934fb5c554f3a2bfb7c33c.png)
. Show that the angle
![\measuredangle KCL](/media/m/7/c/6/7c6cbfe69b4536857d1c1ed9b8a719d7.png)
is independent of the line
![g](/media/m/9/5/8/958b2ae8c90cadb8c953ce50efb9c02a.png)
.
%V0
Let $ABCD$ be a parallelogram. A variable line $g$ through the vertex $A$ intersects the rays $BC$ and $DC$ at the points $X$ and $Y$, respectively. Let $K$ and $L$ be the $A$-excenters of the triangles $ABX$ and $ADY$. Show that the angle $\measuredangle KCL$ is independent of the line $g$.
Izvor: Međunarodna matematička olimpijada, shortlist 2005