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## Main Question or Discussion Point

This is a fairly standard maximization problem in calculus, but I was wondering if anybody could help me come up with a nice geometric solution. It seems like it should be possible to make an argument based on symmetry, but I haven't quite been able to work it out yet. Note, I have already solved this problem using calculus and/or trigonometry, but I'm interested in a geometric solution. Help is greatly appreciated.

The Right Triangle Problem

The sum of the lengths of the two legs of a right triangle is greater

than the length of its hypotenuse. [Let’s agree that the three sided

of a triangle must be genuine line segments, not single points.]

Considering only right triangles with hypotenuse of length 1, what is

the largest the difference between the sum of the lengths of the two

sides and the length of the hypotenuse can be?

The Right Triangle Problem

The sum of the lengths of the two legs of a right triangle is greater

than the length of its hypotenuse. [Let’s agree that the three sided

of a triangle must be genuine line segments, not single points.]

Considering only right triangles with hypotenuse of length 1, what is

the largest the difference between the sum of the lengths of the two

sides and the length of the hypotenuse can be?