<s>[INST] Find side of an equilateral triangle inscribed in a rhombus The lengths of the diagonals of a rhombus are 6 and 8. An equilateral triangle inscribed in this
rhombus has one vertex at an end-point of the shorter diagonal and one side parallel to the longer
diagonal. Determine the length of a side of this triangle. 
Express your answer in the form $k\left(4 \sqrt{3} − 3\right)$ where k is a vulgar fraction.
 [/INST] 
Given the picture, let $x$ be the side of the equilateral triangle.
We have:
$$ 6 = \frac{x}{2}\cot\arctan\frac{4}{3}+\frac{x}{2}\cot\frac{\pi}{6},$$
or:
$$ 6 = \frac{3x}{8}+\frac{\sqrt{3}\,x}{2},$$
so:
$$ 48 = x(4\sqrt{3}+3) $$
and:
$$ 48(4\sqrt{3}-3) = 39 x,$$
so $k=\color{red}{\frac{16}{13}}$.
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