Puzzle of the Week #419 - Minimum Triangle

This right-angled triangle is shown with its inscribed circle (incircle). The distances a and b make up the distance from each end of the base to the tangent point of the incircle. c is the height of the triangle, perpendicular to the base.

If I tell you that a, b and c are all whole numbers, and all different, what is the minimum sum of a, b and c?