Hopping a Triangular Path

Geometry Level 4

Kelvin the Frog lives in a triangle ABCABC with side lengths 4, 5 and 6. One day he starts at some point on side ABAB of the triangle, hops in a straight line to some point on side BCBC of the triangle, hops in a straight line to some point on side CACA of the triangle, and finally hops back to his original position on side ABAB of the triangle. The smallest distance Kelvin could have hopped is mn\frac{m}{n} for relatively prime positive integers mm and nn. What is m+nm+n?

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