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In rectangle $ABCD$, $AB = 6$ and $BC = 8$. Equilateral triangles $ADE$ and $DCF$ are drawn on the exterior of the rectangle. If the area of $ABEF$ is $a \sqrt 3 + b$, find the ordered pair of rational numbers $(a, b)$.

What I did: Obviously it's possible to do this analytically with coordinate geometry, but that gets really messy because you get this:

$$ BE^2 = (6 + 4 \sqrt 3)^2 + 4^2\\ EF^2 = (4 \sqrt 3 + 3)^2 + (4 + 3 \sqrt 3)^2\\ FB^2 = 3^2 + (3 \sqrt 3 + 8)^2 $$

And then you have to expand those expressions, take the square root of all of those sides, and then apply Heron's formula which will add prohibitively many radicals; yuck. Is there a simpler/more elegant way to do this? I'm even willing to accept another analytic/coordinate solution so as long as it isn't unreasonably bashy like this one.

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Hint without words:

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