Pythagorean triples that sum to 60

[Mr Davis 97]

I'm trying to find pythagorean triples that sum to 60. Just from memory, I kow that 3-4-5 and 5-12-13, scaled to some factor, will give triples that sum to 60. These seem to be the only ones that sum to 60, but how can I be sure that there aren't more triples that sum to 60?

 

[fresh_42]

What do you mean by "sum up"? I don't see the $60$.
You can find all by $x = u^2 - v^2 \; , \; y = 2uv \; , \; z = u^2 + v^2$.

 

[Infrared]

Let $x,y,z$ be the side lengths of a triangle you describe, so that $x^2+y^2=z^2$ and $x+y+z=60$. You can solve for $z$ in the last equation and plug it into the first, getting $x^2+y^2=(60-(x+y))^2=3600-120(x+y)+x^2+2xy+y^2$. This gives the equation $60x+60y-xy=1800$, which you can rearrange into $(60-x)(60-y)=1800$. Now you just need to look for positive factor pairs of $1800$ with both factors smaller than $60$ (since $0<x,y<60$).