Look at it in another way

Geometry Level 3

In the diagram, $\triangle{ABC}, \triangle{CDE}, \triangle{EFG}$ and $\triangle{HGI}$ are congruent isosceles triangles. $B,C,E,G$ and $I$ are collinear. Suppose that $\overline{AC}$ and $\overline{BH}$ meet at $M$ and $\overline{FG}$ and $\overline{BH}$ meet at $N$ . Given $[HGN]=10$ , then the value of $[BMC]$ is $\frac{m}{n}$ for relatively prime $m$ and $n$ . Find $m+n$ .

$[ABC]$ denotes the area of figure $ABC$ .