Equal-distance arrangements

Arrange pieces so that chosen distances are equal, for example three pieces mutually the same distance apart, and enumerate the valid configurations. Distance on the grid can be measured in different ways: straight-line (Euclidean) distance uses $\sqrt{(\Delta x)^2 + (\Delta y)^2}$, while king-move distance counts steps. Which condition you enforce depends on which distance you mean.

Enumeration means listing every arrangement that satisfies the equal-distance condition and arguing you have them all, usually by fixing one piece and using symmetry so rotations and reflections are not recounted.