MEMO 2010 ekipno problem 5
Dodao/la:
arhiva28. travnja 2012. The incircle of the triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
touches the sides
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
,
![CA](/media/m/a/a/e/aaec86bc003cfdb64d54116a4cabd387.png)
, and
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
in the points
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
,
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
and
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
, respectively. Let
![K](/media/m/e/1/e/e1ed1943d69f4d6a840e99c7bd199930.png)
be the point symmetric to
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
with respect to the incenter. The lines
![DE](/media/m/a/c/d/acdf3f4d3c794d9a897484e9d216f5ec.png)
and
![FK](/media/m/1/8/5/185a554a2095520f3032542c84f6aa69.png)
intersect at
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
. Prove that
![AS](/media/m/c/0/4/c04e590443646cf2cdc6073f16b39832.png)
is parallel to
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
.
%V0
The incircle of the triangle $ABC$ touches the sides $BC$, $CA$, and $AB$ in the points $D$, $E$ and $F$, respectively. Let $K$ be the point symmetric to $D$ with respect to the incenter. The lines $DE$ and $FK$ intersect at $S$. Prove that $AS$ is parallel to $BC$.
Izvor: Srednjoeuropska matematička olimpijada 2010, ekipno natjecanje, problem 5