Q:

finding a triangle where the square root of two sides is equal to the square root of third side and all side have natural lengths

Is there a method to prove there exists a triangle such that sum of square root of 2 sides is equal to square root of other side which was quoted by 'the Scarecrow' in the Wizard of Oz and more Mathematically.
I was not able to think of an easy proof for do this so I wrote a python code for finding it(Please if you have suggestion leave here in the stack overflow thread)
But it may time a long time with my code if it exist or it will keep on run infinitely because there is no triangle with sides of length in natural numbers.
So, is there a way to get a proof that exists or am I wasting my time?
if there are any mistake in the body and title please edit it or comment it.

A:

If the sides of the triangle are positive numbers $a,b,c$ then you want to have $$ \sqrt a + \sqrt b = \sqrt c$$
Square both sides to get $$a+b + 2\sqrt ab = c$$ 
That is the side $c$ is greater than the sum of the other two sides.
This is contradictory to the well known triangle ineqauality which states that in any triangle, each side is less than or equal to the sum of the other two sides.

