Rational distances are distances whose values are rational numbers. A rational distance set is a subset of Euclidean space for which every pair of points has a rational distance.
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, triangle 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 shown that there are infinitely many triangles with rational sides (Heronian triangles) with two rational triangle medians (Guy 1994, p. 188).
Qiu (2026) constructed, for every integer 
, a denumerable set
in 
with all pairwise distances rational and with
no 
points lying in a common hyperplane.
This is affine general position. For odd 
, the set can
additionally be chosen so that no 
points lie on a common hypersphere.
In particular, in three dimensions, every four points are
noncoplanar and no five lie on a common sphere.
Clearing the distance denominators in any finite subset gives arbitrarily large finite configurations with
integer distances and the same position properties (Qiu
2026). This scaling argument applies to each finite subset
separately. For 
,
the result does not exclude four concyclic points.
