This is the entire problem. As a point has no width or height, let's say there is an infinite number of them. Take a table of size 3x3 and try to insert red or blue points so that no three of them form a right isosceles triangle. You can't do that, and this is what I have to prove.
If there is an infinite number of them than any possible shape will be made (that's less than a 3x3 box) So a right isosceles triangle will inevitably occur.
It is like the monkey's on a keyboard. Given an infinite amount of time, they will write the completed works of Shakespeare.