Distances in a Right Triangle

Geometry Level 5

\(ABC\) is a right triangle with \(\angle ABC = 90^\circ\) with \(AB = 30\sqrt{3}\) and \(BC = 30\). \(D\) is a point on segment \(BC\) such that \(AD\) is the median. \(E\) is a point on segment \(AC\) such that \(BE\) is perpendicular to \(AC\). \(AD\) and \(BE\) intersect at \(F\). If \(EF = \frac{a\sqrt{b}}{c}\), where \(a\) and \(c\) are positive, coprime integers and \(b\) is not divisible by the square of any prime, what is the value of \(a + b + c\)?

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