What is the locus of points the difference of whose distances from two points being constant?

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Given two points on a plane, the locus of points with a constant distance difference is a hyperbola.

What happens if there are three points on the plane? Concretely, if there are three points A, B, and C, I'm looking for locus of all points X such that mod(d(X, A) - d(X, B)) = mod(d(X, B) - d(X, C)) = mod(d(X, C) - d(X, A)) = constant.

Intuitively, it seems like no such points exist as the three hyperbola don't intersect at same points. How to make this formal?