Distances in a Right Triangle

Geometry Level 5

ABCABC is a right triangle with ABC=90\angle ABC = 90^\circ with AB=303AB = 30\sqrt{3} and BC=30BC = 30. DD is a point on segment BCBC such that ADAD is the median. EE is a point on segment ACAC such that BEBE is perpendicular to ACAC. ADAD and BEBE intersect at FF. If EF=abcEF = \frac{a\sqrt{b}}{c}, where aa and cc are positive, coprime integers and bb is not divisible by the square of any prime, what is the value of a+b+ca + b + c?

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