TITLE: Finding the shaded area in a triangle
QUESTION [17 upvotes]: Here is the diagram:

I only know that the middle segment is a median of the big triangle. But nothing else.

REPLY [3 votes]: Proof without words:
$\hspace{3cm}$
$$\frac gh=\frac{2}{\sqrt{h^2+9}}=\frac{\sqrt{h^2+1}}{\sqrt{h^2+36}} \Rightarrow h^4+6h^2-135=0 \Rightarrow h=3.$$
$$S=\frac12(3+1)3=6.$$