The Wheel that leaves Pythagorean Triads in its Wake

A Pythagorean triad $(x,y,u)$ consists of positive integers $x,y,u$ such that $x^2+y^2=u^2$. Geometrically, the integers represent the lengths of the sides of a right-angled-triangle with the hypotenuse $u$. It immediately follows from Pythagoras' Theorem that the remaining sides cannot have equal length ($\sqrt{2}$ is irrational). Thus, without loss of generality, we suppose that $x<y$ in the following. The $(3,4,5)$ Pythagorean triad is well known and it is obvious that if $(x,y,u)$ is a Pythagorean triad then so is $(kx, ky, ku)$ for any integer $k>0$. (The case $k=1$ is a primitive Pythagorean triad.) So, $(6,8,10)$ is a Pythagorean triad and infinitely many more can be constructed. Geometrically these are similar triangles, but $(5, 12, 13)$ is also a Pythagorean triad and the triangle with sides $(5, 12, 13)$ is not similar to the triangle with sides $(3, 4, 5)$.