It is possible to find six points in the plane, no three on a line and no four on a circle
(i.e., none of which are collinear or concyclic),
such that all the mutual distances are rational.
An example is illustrated by Guy (1994, p. 185).
Guy, R. K. "Six General Points at Rational Distances" and "Triangles with Integer Sides, Medians, and Area." §D20 and D21
in Unsolved
Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 185-190,
1994.