IMO Shortlist 2009 problem G8
Dodao/la:
arhiva2. travnja 2012. Let
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
be a circumscribed quadrilateral. Let
![g](/media/m/9/5/8/958b2ae8c90cadb8c953ce50efb9c02a.png)
be a line through
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
which meets the segment
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
in
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
and the line
![CD](/media/m/8/9/5/895081147290365ccae028796608097d.png)
in
![N](/media/m/f/1/9/f19700f291b1f2255b011c11d686a4cd.png)
. Denote by
![I_1](/media/m/4/1/6/416fe38d243ecd6d66747e4d88b6518d.png)
,
![I_2](/media/m/c/b/0/cb0739259063636b82f7b6b3e9dd3de0.png)
and
![I_3](/media/m/9/d/8/9d8b7d345598e5f924b2b2d7b253b2aa.png)
the incenters of
![\triangle ABM](/media/m/a/4/6/a46087d896fc7ad5906475cab3c43ada.png)
,
![\triangle MNC](/media/m/1/6/a/16a5eaa817b1619f4fc32dc4656e7630.png)
and
![\triangle NDA](/media/m/7/c/f/7cf1b83e882ea1b64809ff185cbaa509.png)
, respectively. Prove that the orthocenter of
![\triangle I_1I_2I_3](/media/m/5/7/6/57642900758363e6b9d6f858ed833308.png)
lies on
![g](/media/m/9/5/8/958b2ae8c90cadb8c953ce50efb9c02a.png)
.
Proposed by Nikolay Beluhov, Bulgaria
%V0
Let $ABCD$ be a circumscribed quadrilateral. Let $g$ be a line through $A$ which meets the segment $BC$ in $M$ and the line $CD$ in $N$. Denote by $I_1$, $I_2$ and $I_3$ the incenters of $\triangle ABM$, $\triangle MNC$ and $\triangle NDA$, respectively. Prove that the orthocenter of $\triangle I_1I_2I_3$ lies on $g$.
Proposed by Nikolay Beluhov, Bulgaria
Izvor: Međunarodna matematička olimpijada, shortlist 2009