Random Point in a Hexagon

Probability Level 5

A point PP is uniformly chosen inside a regular hexagon of side length 33. For each side of the hexagon a line is drawn from PP to the point on that side which is closest to PP. The probability that the sum of the lengths of these segments is less than or equal to 939\sqrt{3} can be expressed as ab\frac{a}{b} where aa and bb are coprime positive integers. What is the value of a+ba + 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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