Prove that 4 points are concyclic In $\triangle ABC$ it holds $BC = \frac{AB+AC}{2}$ .Let $M$ and $N$ be midpoints of $AB$  and $AC$ ,and let $I$ be the incentre of $\triangle ABC$. Prove that $A,M,I,N$ are concyclic.
I have tried it very hard. But there are some things that i could decipher.
$$\frac{AO}{OH}=\frac{AM}{MB}=\frac{AN}{NC}=1$$. Rest are drawn on the figure.


hint: 
connect $BI,CI$, check $\triangle MIB$  and $\triangle HIB$,what is their relation. then what is the relation between $\angle IMB$ and $\angle IHB$.
similar method, check $\angle INC$ and $\angle IHC $,now you can prove the result.
