  We show that planar embeddable 3-connected CAD graphs are generically
non-soluble. A CAD graph represents a configuration of points on the Euclidean
plane with just enough distance dimensions between them to ensure rigidity.
Formally, a CAD graph is a maximally independent graph, that is, one that
satisfies the vertex-edge count 2v - 3 = e together with a corresponding
inequality for each subgraph. The following main theorem of the paper resolves
a conjecture of Owen in the planar case. Let G be a maximally independent
3-connected planar graph, with more than 3 vertices, together with a realisable
assignment of generic dimensions for the edges which includes a normalised unit
length (base) edge. Then, for any solution configuration for these dimensions
on a plane, with the base edge vertices placed at rational points, not all
coordinates of the vertices lie in a radical extension of the dimension field.
