First of all, Your theorem is correct and so is your proof.
As Sean pointed out, your theorem is very restrictive Since it requires some conditions on where the triangles locate in the plane. You mentioned that we can deal with that by transformations and rotations and so on. Ok, This could work well, but in mathematics, there are some questions to be answered when we face a new theorem.
For example, Are the conditions the theorem requires necessary? are they only sufficient?
are they both?
If the later is the case then this is a great news! this doesn't mean that if the conditions are only sufficient but not necessary, that this is bad, not at all. What I mean is that if we have some sufficient and necessary conditions then this is better.
For your theorem, your conditions are only "sufficient" but not necessarly required. We have already some theorems whose conditions are both, necessary and sufficient.
Having said that, the question is, Do we really need this particular theorem? Are there any problems of importance which can't be solved easily without your theorem? If this is the case, I think your efforts will be appreciated, otherwise, why to take your theorem into consideration at all if we have "better" tools to do everything your theorem does?
I mean, Why to consider this as a theorem not as an exercise in any geometry book? You should recognise that "theorems" in mathematics are the important facts which are essential to understand the mathematical releam So not everything we prove is a "theorem".