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 $|XA - XB| = |XB-XC| = |XC - XA| = constant$.
Intuitively, it seems like no such points exist as the three hyperbola don't intersect at same points. How to make this formal?