IMO Shortlist 2004 problem G8
Dodao/la:
arhiva2. travnja 2012. Given a cyclic quadrilateral
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
, let
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
be the midpoint of the side
![CD](/media/m/8/9/5/895081147290365ccae028796608097d.png)
, and let
![N](/media/m/f/1/9/f19700f291b1f2255b011c11d686a4cd.png)
be a point on the circumcircle of triangle
![ABM](/media/m/1/0/f/10f2e13fa27b4f7846a6ffc4baab3603.png)
. Assume that the point
![N](/media/m/f/1/9/f19700f291b1f2255b011c11d686a4cd.png)
is different from the point
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
and satisfies
![\frac{AN}{BN}=\frac{AM}{BM}](/media/m/f/b/c/fbc4c3c8888735a94790b47cbbf60a77.png)
. Prove that the points
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
,
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
,
![N](/media/m/f/1/9/f19700f291b1f2255b011c11d686a4cd.png)
are collinear, where
![E=AC\cap BD](/media/m/6/6/c/66cbda5511838fb2506bc39bb447b124.png)
and
![F=BC\cap DA](/media/m/d/7/c/d7cb01632615f5e5d6267f34da5459ad.png)
.
%V0
Given a cyclic quadrilateral $ABCD$, let $M$ be the midpoint of the side $CD$, and let $N$ be a point on the circumcircle of triangle $ABM$. Assume that the point $N$ is different from the point $M$ and satisfies $\frac{AN}{BN}=\frac{AM}{BM}$. Prove that the points $E$, $F$, $N$ are collinear, where $E=AC\cap BD$ and $F=BC\cap DA$.
Izvor: Međunarodna matematička olimpijada, shortlist 2004