For plane , there are at least
different distances . The minimum distance can occur only maximum distance
can occur distance can occur as often as
times.
Finally, no set of plane can determine only isosceles
triangles .
See also Distance
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References Honsberger, R. "The Set of Distances Determined by Mathematical
Gems II. Referenced
on Wolfram|Alpha Planar Distance
Cite this as:
Weisstein, Eric W. "Planar Distance."
From MathWorld https://mathworld.wolfram.com/PlanarDistance.html
Subject classifications
points in the plane, there are at least
times, and the maximum distance
can occur 
times. Furthermore, no distance can occur as often as
points in the plane can determine only isosceles
triangles.
Points in the Plane." Ch. 12
in