MEMO 2014 ekipno problem 6
Dodao/la:
arhiva24. rujna 2014. Let the incircle
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
of the triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
touch its side
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
at
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
. Let the line
![AD](/media/m/6/9/6/69672822808d046d0e94ab2fa7f2dc80.png)
intersect
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
at
![L \neq D](/media/m/7/b/2/7b20dba0ee80cd595b3bce8f2ff7243c.png)
and denote the excentre of
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
opposite to
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
by
![K](/media/m/e/1/e/e1ed1943d69f4d6a840e99c7bd199930.png)
. Let
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
and
![N](/media/m/f/1/9/f19700f291b1f2255b011c11d686a4cd.png)
be the midpoints of
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
and
![KM](/media/m/0/1/c/01ca9badbb81b31fdf7b3e19f0f0c6c6.png)
respectively.
Prove that the points
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
,
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
,
![N](/media/m/f/1/9/f19700f291b1f2255b011c11d686a4cd.png)
, and
![L](/media/m/f/c/1/fc1ae4eb78da7d1352cbf1f8217ab286.png)
are concyclic.
%V0
Let the incircle $k$ of the triangle $ABC$ touch its side $BC$ at $D$. Let the line $AD$ intersect $k$ at $L \neq D$ and denote the excentre of $ABC$ opposite to $A$ by $K$. Let $M$ and $N$ be the midpoints of $BC$ and $KM$ respectively.
Prove that the points $B$, $C$, $N$, and $L$ are concyclic.
Izvor: Srednjoeuropska matematička olimpijada 2014, ekipno natjecanje, problem 6