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?

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