Let be a circumscribed quadrilateral. Let be a line through which meets the segment in and the line in . Denote by , and the incenters of , and , respectively. Prove that the orthocenter of lies on .
Proposed by Nikolay Beluhov, Bulgaria
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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
Školjka 
be a circumscribed quadrilateral. Let 
be a line through 
which meets the segment 
in 
and the line 
in 
. Denote by 
, 
and 
the incenters of 
, 
and 
, respectively. Prove that the orthocenter of 
lies on