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 \(ABC\).

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