There Are Too Many Things!

Geometry Level 5

Let \(ABC\) be a triangle with incircle \(\omega\). Let \(D\), \(E\) and \(F\) be the points of tangency of \(\omega\) with \(AB\), \(BC\) and \(CA\). Let \(G\) be the intersection of \(DF\) and \(BC\) and \(H\) be the intersection of \(DE\) and \(AC\) (\(G\), \(H\), \(E\) and \(F\) are on the same side of \(AB\)). Let \(M\) be the midpoint of \(FH\) and \(N\) be the midpoint of \(EG\).

If the area of \(ABMN\) is 100 and the area of \(CMN\) is 4, find the area of \(EFGH\).

Bonus: Can you generalise this?

Kudos to Xuming Liang for helping me solve this question.

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