Computational Geometry has a technique for avoiding all of the complex cases where point are colinear, coplanar, cocircular, etc.. I’ve forgotten the techniques’ name, but coordinates are integers plus an epsilon term. As the epsilon goes to zero, you get the original statement of the problem.
The epsilon terms are used for breaking ties. If you are deciding if a point is to the right or left of a line, you can use the factors on epsilon terms to decide. If the limit of the point for epsilon close to but greater than 0 is right or left, you use that answer.
You can use multiple epsilon terms, one for each dimension and claim that the x-dimension approaches zero faster than the y or z-dimension, etc. It makes it all work as integer linear algebra with simple greater than or less than on scalar result.
The epsilon terms are used for breaking ties. If you are deciding if a point is to the right or left of a line, you can use the factors on epsilon terms to decide. If the limit of the point for epsilon close to but greater than 0 is right or left, you use that answer.
You can use multiple epsilon terms, one for each dimension and claim that the x-dimension approaches zero faster than the y or z-dimension, etc. It makes it all work as integer linear algebra with simple greater than or less than on scalar result.