Yes, it does actually. I set out to do a completely algebraic proof that a->b->c->a (a the law of sines, b the PT, c the law of cosines). All of which assumes the Euclidean plane (ie triangles angles sum to 180).

But I have since taken two extra rather important points from my paper after writing it.

1) The paper is of little value since all three are more easily derived "from scratch." The fact that they circularly imply each other is not a big surprise, as Alex said, as this quite often happens in mathematics. There is also still this unanswered question as to whether one has license to use sin(A+B) = sin(C) = 1 to prove the PT. I set out to defend this in extending the definition of sine to angles that are not acute (90 in particular). So I suppose one could say that instead of proving the PT, I just picked out a special case of the law of sines which happens to lead to the PT (again no surprise).

2) The PT is not really necessary in this at all. Drawing the three altitudes in a triangle both leads to the law of sines and the law of cosines, without recourse to the PT (see my other post where I have linked to my PWW of the cosine law independent of the PT). That same diagram (without the exterior rectangles and square) is also what is used to derive the law of sines.

And so the law of sines and the law of cosines both stem from drawing altitudes in a triangle. The PT is just a special case where the orthocenter is also a vertex of an angle.

If you follow the other thread, you will see that I am attempting to create (or convince Alex to create) a droodle that illustrates this fact from my aforementioned PWW.

Thanks for your insight.