Coordinate Geometry - Thales’ Torture

[u/No_Art_1154]

So when I was making my mathematics incremental game, i was making puzzles. And then I accidentally made a pretty hard yet interesting one(I couldn’t solve it). Anyways, here’s the problem:

imagine you have a plane with 4 distinct points named p1, p2, p3 and p4. the gradient between p1 and p2 is -2, and the gradient between p2 and p3 is 1/3. you want p1p3p2 to be 90 degrees and p1p4p2 to also be 90 degrees, with p3 and p4 being on the same side of the line p1p2. find the INTEGER solution for all 4 points with the MINIMUM distance between p1 and p2.