TITLE: Are angles between points enough to decide the realizability?
QUESTION [8 upvotes]: Let n points in the plane be given whose coordinates we don't know.
Assume, however, that for any triple of the points we know the angle.
Question: Can we decide whether the n points are realizable in the plane, i.e. can we determine whether there is an assignment of coordinates to the points such that the angular constraints are satisfied?

REPLY [4 votes]: To answer Joonas's comment in the last paragraph: yes, it is possible to check the realizability without constructing the whole realization. It is sufficient to test whether every 5-tuple of points 
is realizable. (For simplicity, I am also assuming general position, that is, no three points on a line.)
To see this, note that when a new point is being added in Joonas's algorithm, its location is determined by an arbitrary triple of points already drawn. The new point can be drawn if all the triples agree on the same location. If two triples disagree, then there are also two triples sharing two points that disagree. Together with the new point, this makes a 5-tuple that is not realizable.
By some finite case analysis of all 5-tuples, it can be probably shown that testing 4-tuples is also sufficient.