IMO Shortlist 2008 problem G1
Let
be the orthocenter of an acute-angled triangle
. The circle
centered at the midpoint of
and passing through
intersects the sideline
at points
and
. Similarly, define the points
,
,
and
.
Prove that six points
,
,
,
,
and
are concyclic.
Author: Andrey Gavrilyuk, Russia
%V0
Let $H$ be the orthocenter of an acute-angled triangle $ABC$. The circle $\Gamma_{A}$ centered at the midpoint of $BC$ and passing through $H$ intersects the sideline $BC$ at points $A_{1}$ and $A_{2}$. Similarly, define the points $B_{1}$, $B_{2}$, $C_{1}$ and $C_{2}$.
Prove that six points $A_{1}$ , $A_{2}$, $B_{1}$, $B_{2}$, $C_{1}$ and $C_{2}$ are concyclic.
Author: Andrey Gavrilyuk, Russia
Source: Međunarodna matematička olimpijada, shortlist 2008
Školjka 
