Mathematics

The key is to set up a coordinate system with the right angle at the origin and legs along the axes, then use the constraint that a point on the hypotenuse maintains fixed perpendicular distances $a$ and $b$ to those legs. This transforms the problem into minimizing the hypotenuse length $c = x + y$ subject to the geometric constraint that the point's distances to the legs equal $a$ and $b$—which forces a relationship between the leg lengths. Calculus or the AM-GM inequality applied to the reciprocals of the legs will reveal that the minimum occurs at a specific ratio, leading to the elegant fractional-power form.

The key is to set up a coordinate system with the right angle at the origin and legs along the axes, then use the constraint that a point on the hypotenuse maintains fixed perpendicular distances $a$ and $b$ to those legs. This transforms the problem into minimizing the hypotenuse length $c = x + y$ subject to the geometric constraint that the point's distances to the legs equal $a$ and $b$—which forces a relationship between the leg lengths. Calculus or the AM-GM inequality applied to the reciprocals of the legs will reveal that the minimum occurs at a specific ratio, leading to the elegant fractional-power form.