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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).
It is not known if a Triangle with Integer sides, Medians, and Area exists (although there are incorrect Proofs of the impossibility in the literature). However, R. L. Rathbun, A. Kemnitz, and R. H. Buchholz have showed that there are infinitely many triangles with Rational sides (Heronian Triangles) with two Rational Medians (Guy 1994, p. 188).
See also Collinear, Concyclic, Cyclic Quadrilateral, Equilateral Triangle, Euler Brick, Heronian Triangle, Rational Quadrilateral, Rational Triangle, Square, Triangle
References
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.