By a quick search after reading your first paragraph:

In the rectangle from (0,0) to (56,63), the point (20,48) is at integer distances from all the corners. Going clockwise from the origin, the distances are 52, 25, 39, and 60.

By a quick search after reading your first paragraph:

In the rectangle from $(0,0)$ to $(56,63)$, the point $(20,48)$ is at integer distances from all the corners. Going clockwise from the origin, the distances are $52, 25, 39$, and $60$.

Edit. Corresponding Pythagorean triples (in the same order) are $20^2+48^2=52^2$, $20^2+15^2=25^2$, $36^2+15^2=39^2$, and $36^2+48^2=60^2$.