# Random Point in a Hexagon

A point $$P$$ is uniformly chosen inside a regular hexagon of side length $$3$$. For each side of the hexagon a line is drawn from $$P$$ to the point on that side which is closest to $$P$$. The probability that the sum of the lengths of these segments is less than or equal to $$9\sqrt{3}$$ can be expressed as $$\frac{a}{b}$$ where $$a$$ and $$b$$ are coprime positive integers. What is the value of $$a + b$$?

Details and assumptions

The side of the hexagon is a line segment, not a line.

Note that the 6 closest points are always distinct, hence we will have 6 distinct line segments.

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