@Carl Love 

That really helpped.

Yes I agree that factorization is the wrong term for what I am doing - maybe formulating?

Following on, I tried a variation on the guided simplify operation:

Subs5 := {(p1-p2)*(p1-p3)*(p2-p3) = dp121323, (p1-p4)*(p2-p5)*(p3-p6) = dp142536, (p1-p4)*(p2-p6)*(p3-p5) = dp142635, (p1-p5)*(p2-p4)*(p3-p6) = dp152436, (p1-p5)*(p2-p6)*(p3-p4) = dp152634, (p1-p6)*(p2-p4)*(p3-p5) = dp162435, (p1-p6)*(p2-p5)*(p3-p4) = dp162534, (p4-p5)*(p4-p6)*(p5-p6) = dp454656}

simplify(T2, Subs5)

This seemed to hang the tool, I left it overnight, nothing in the morening. Do you see anything inherently wrong with this approach?

@Carl Love 

Not expecting T2 to be divisible by p1-p2. I agree it clearly cannot. What I am asking is why cant Maple factor b^2-4ac into terms involving (Px-Py). I have proven that b^2-4ac = (p1-p2)^2+(p3-p4)^2 - 2*((p1-p3)*(p2-p4)+(p1-p4)*(p2-p3)), simply by the fact that T1-T2=0. Is there some way to steer the FACTOR command (or some other command) so that Maple can find the aforementiond factorization: terms involving only (Px-Py) with x not equal y. With this knowledge in hand, I would like to try the case in the first file I downloaded. Thanks.

@Carl Love 

if you are referring to (19) in file divide.mw, then b&c are functions of Px. If you are referring to (24) then terms dpxy are functions of Px & Py. If this doesnt clarify, then I dont understand your update statement.