Hopping a Triangular Path

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 $m+n$?