Three circles, one triangle and a pink area

Geometry Level 4

We have an equilateral triangle of side \( a \) (its height will be labelled as \(h \)). If each side's midpoint is a circumcentre for a circle of radius \( \frac{1}{3}h \), find the triangle's area that is not overlapped by any of the circles (pink area on the image below).

If \( m, n \) are both positive integers, then the area can be presented as: \( \frac{m\sqrt{3} - \pi}{n} a^{2} \)

Submit answer as: \( |m - n| \)

Note: keep in mind that this image is done in paint, therefore it is very inaccurate.


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