<s>[INST] Finding the shaded area in a triangle Here is the diagram:

I only know that the middle segment is a median of the big triangle. But nothing else.
 [/INST] Let $h$ be the height of the triangles.  Then the area of the large triangle is $$\Delta_{\text{ large}} = \frac{1}{2}\times 6\times \sqrt{h^2+36} \times \sin a$$ and the area of the white triangle is $$\Delta_{\text{ white}} =\frac{1}{2} \times\sqrt{h^2+1}\times\sqrt{h^2+9}\times\sin a$$
But since the large triangle has the same height as the white triangle, but three times its base, we have $\Delta_{\text{ large}} = 3 \Delta_{\text{white}}$.  So
$$ 2\sqrt{h^2+36} = \sqrt{h^2+1}\sqrt{h^2+9}$$
Squaring both sides and simplifying gives
$$h^4+10h^2+9=4h^2+144$$
$$\Rightarrow (h^2+15)(h^2-9) = 0$$
So $h=3$, and the shaded area is $6$.
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