Lego triangles and Pythagorean triples

Usually Lego Technic models are based on square patterns of holes which in turn are based on the Lego length unit “1 stud”.

The problem is when you want to attach a leaning liftarm to your (square) construction, how do you find the correct holes to attach it to? Usually the leaning liftarm “almost” fits.

I translated the issue into a mathematical problem. The base is of course a right triangle with the condition that all sides must be an integer. The phenomenon is called Pythagorean triples. (google it). I found the following triples especially interesting for the context of Lego.

3-4-5
6-8-10
5-12-13

Then I realized that Lego also “supports” “half stud” constructions which also enable the following combinations:

1,5-2-2,5
2,5-6-6,5

Note that the numbers above corresponds to the length between the first and last hole in a liftarm. To translate the length into the number of holes of the liftarms you have to add one (1). Below you can see the examples of the variations.