It's possible to divide a triangle into smaller triangles such that no edge lengths are shared. Alternately, no two internal triangles share two vertices. The top three are the known simplest solutions. The bottom three were found via doodling.
The triangles in the first solution could be subdivided the same way, so another criterion is added: no subdivided internal triangles are allowed.
What are some other solutions with 7 to 18 internal triangles?
New solutions might be found by analyzing Schlegel diagrams of planar graphs, and picking three vertices of each face to be a triangle. If no two triangles share two faces, and if excised facial vertices can be moved between triangular vertices, then it's a potential solution. For example, this graph leads to the upper right solution, with triangles 234, 478, 456, 138, 125, 167.
One more solution, with nine of the triangles having area 1.