Hopping a Triangular Path

Geometry Level 5

Kelvin the Frog lives in a triangle \(ABC\) with side lengths 4, 5 and 6. One day he starts at some point on side \(AB\) of the triangle, hops in a straight line to some point on side \(BC\) of the triangle, hops in a straight line to some point on side \(CA\) of the triangle, and finally hops back to his original position on side \(AB\) of the triangle. The smallest distance Kelvin could have hopped is \(\frac{m}{n}\) for relatively prime positive integers \(m\) and \(n\). What is \(m+n\)?

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