IMO Shortlist 2000 problem G3
Dodao/la:
arhiva2. travnja 2012. Let
![O](/media/m/9/6/0/9601b72f603fa5d15addab9937462949.png)
be the circumcenter and
![H](/media/m/4/c/0/4c0872a89da410a25f00b86366efece7.png)
the orthocenter of an acute triangle
![ABC.](/media/m/c/b/7/cb77700b4adade65e440645391a8d2ad.png)
Show that there exist points
![D, E,](/media/m/6/9/8/698de974cb2fd8b696db9792f20aaf54.png)
and
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
on sides
![BC,CA,](/media/m/1/0/6/1062d6c9d8a9816f90ff13930b60a0c1.png)
and
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
respectively such that
![OD + DH = OE +EH = OF +FH](/media/m/a/6/7/a674e435ee7d74b32e3127a76f649bc8.png)
and the lines
![AD, BE,](/media/m/f/3/3/f33064728d6bd991ad190fcb4e6da732.png)
and
![CF](/media/m/6/7/0/670c216bc8a05762a60542376587c5fc.png)
are concurrent.
%V0
Let $O$ be the circumcenter and $H$ the orthocenter of an acute triangle $ABC.$ Show that there exist points $D, E,$ and $F$ on sides $BC,CA,$ and $AB$ respectively such that $$OD + DH = OE +EH = OF +FH$$ and the lines $AD, BE,$ and $CF$ are concurrent.
Izvor: Međunarodna matematička olimpijada, shortlist 2000