IMO Shortlist 1986 problem 14
Dodao/la:
arhiva2. travnja 2012. The circle inscribed in a triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
touches the sides
![BC,CA,AB](/media/m/9/8/c/98c204ffa459114826231180fce7ec09.png)
in
![D,E, F](/media/m/2/1/4/214730f54252a42c646e4400a3377bf3.png)
, respectively, and
![X, Y,Z](/media/m/e/e/4/ee4449852a793f7deadce6fa244d0a5b.png)
are the midpoints of
![EF, FD,DE](/media/m/7/6/6/766b1c7c859b99631c8b92bf31ddabe8.png)
, respectively. Prove that the centers of the inscribed circle and of the circles around
![XYZ](/media/m/1/3/d/13dab5022dd1d33f3d299852f2f54cfb.png)
and
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
are collinear.
%V0
The circle inscribed in a triangle $ABC$ touches the sides $BC,CA,AB$ in $D,E, F$, respectively, and $X, Y,Z$ are the midpoints of $EF, FD,DE$, respectively. Prove that the centers of the inscribed circle and of the circles around $XYZ$ and $ABC$ are collinear.
Izvor: Međunarodna matematička olimpijada, shortlist 1986