I’m learning about descriptive geometry, which uses compass/straightedge techniques to do some nifty things like figure out intersections of cones with cones and find the shortest distance between solids in space. I’m trying to construct a problem for myself, and stumbled into what seems like a deeper question.

I’d like to construct a triangle ABC in 3D space, where the x,y, and z coordinates of each corner are unique integers - no two x the same, no two y, no two z - but also each side length AB, BC, CA is a unique integer. It’s easy to make one edge: use two Pythagorean triples that share one side length to get the diagonal of a prism. So points (0,0,0) and (3,4,12) give you a diagonal of length 13. It seems to me that you could extend this to two more prisms somehow to get the third point of the triangle, but I got lost in a bunch of simultaneous equations.

Is such a triangle possible? Does it have a name? If three different integer side lengths aren’t possible, how about two?

ETA: put another way, is there a way to rotate an integer triangle in space such that the vertices are on integer coordinates?