It only just occurred to me that the geometric constructions we've seen so far for doubly planar charts (no rationality requirement) all build families that preserve a single triangle, and provide alternate placements for the fourth point. Fred's various examples -- integers with the collinearity blemish and generic quartics -- both do likewise. Is this necessary? That is, given the six distances, can you at least identify three of them that must form a triangle, or might there be two planar charts with no triangles in common? (It's possible that someone has posted an example I've missed which already settles this question; I haven't tried to do an exhaustive literature search :-). --Michael -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen.