MEMO 2009 ekipno problem 6
Dodao/la:
arhiva28. travnja 2012. Suppose that
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
is a cyclic quadrilateral and
![CD=DA](/media/m/e/6/3/e63ffb6ca86f96f22304798c3add144e.png)
. Points
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
and
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
belong to the segments
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
and
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
respectively, and
![\angle ADC=2\angle EDF](/media/m/b/3/8/b38e00bbca1af3d434cc11087b577c8b.png)
. Segments
![DK](/media/m/8/e/f/8ef301ce1b0183492d3e43cca080130c.png)
and
![DM](/media/m/b/2/8/b28f740f871a84995535c1b1d29b160e.png)
are height and median of triangle
![DEF](/media/m/8/a/7/8a759311e6e14a716e429d1f5419dde9.png)
, respectively.
![L](/media/m/f/c/1/fc1ae4eb78da7d1352cbf1f8217ab286.png)
is the point symmetric to
![K](/media/m/e/1/e/e1ed1943d69f4d6a840e99c7bd199930.png)
with respect to
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
. Prove that the lines
![DM](/media/m/b/2/8/b28f740f871a84995535c1b1d29b160e.png)
and
![BL](/media/m/d/6/0/d60f78c5ee392997e29360a340fd3fba.png)
are parallel.
%V0
Suppose that $ABCD$ is a cyclic quadrilateral and $CD=DA$. Points $E$ and $F$ belong to the segments $AB$ and $BC$ respectively, and $\angle ADC=2\angle EDF$. Segments $DK$ and $DM$ are height and median of triangle $DEF$, respectively. $L$ is the point symmetric to $K$ with respect to $M$. Prove that the lines $DM$ and $BL$ are parallel.
Izvor: Srednjoeuropska matematička olimpijada 2009, ekipno natjecanje, problem 6