First, we need a definition of "adjacent plane figures". If we take "plane figure" to mean a closed plane curve (the boundary) along with its interior, then one definition might be that two plane figures are adjacent when (a) the intersection of their boundaries is a curve but not a point; (b) their interiors are disjoint. For polygons (boundaries consist of line segments) any subtle topological issues disappear, so this will certainly do for triangles.
Given this, then I think it's probably not too hard to prove this hypothesis:
Theorem: Triangles ABC and DEF are adjacent iff all of these hold:
(a) At least one side of ABC (say AB) and one side of DEF (say DE) are identical as lines (not line segments).
(b) At least one of A and B is strictly between D and E, or at least one of D and E is strictly between A and B.
(c) C and F are on different sides of line AB (which is the same as line DE).
The nice thing about this theorem, if true, is that it should not be hard to come up with a simple algorithm to test for adjacency in the Cartesian plane. I suspect the image processing folks have done this and much much more along the same lines.
Here are some of the trickier cases...