I have the following Mathematica result under the assumption that {ax, bx, px, ay, by, py} \[Element] Reals
, and after refinement by Simplify
:
ConditionalExpression[-((
ay/Sqrt[(ax - px)^2 + (ay - py)^2] - by/
Sqrt[(bx - px)^2 + (by -
py)^2] + (-(1/Sqrt[(ax - px)^2 + (ay - py)^2]) + 1/
Sqrt[(bx - px)^2 + (by - py)^2]) py)/(
ay (bx - px) + by px - bx py + ax (-by + py))),
Im[((ax - bx) (ax - px) + (ay - by) (ay - py) -
I Abs[ay bx - ax by - ay px + by px + ax py - bx py])/((ax -
bx)^2 + (ay - by)^2)] < 0 ||
Re[((ax - bx) (ax - px) + (ay - by) (ay - py) -
I Abs[ay bx - ax by - ay px + by px + ax py - bx py])/((ax -
bx)^2 + (ay - by)^2)] < 0 ||
Im[((ax - bx) (ax - px) + (ay - by) (ay - py) -
I Abs[ay bx - ax by - ay px + by px + ax py - bx py])/((ax -
bx)^2 + (ay - by)^2)] > 0 ||
Re[((ax - bx) (ax - px) + (ay - by) (ay - py) -
I Abs[ay bx - ax by - ay px + by px + ax py - bx py])/((ax -
bx)^2 + (ay - by)^2)] > 1]
Pictorially, it's this:
If you consider the first part of the OR, can you confirm that the Im[...] < 0
is nothing more than the stuff in the absolute value brackets (Abs[ay bx - ax by - ay px + by px + ax py - bx py]
) being greater than zero?
Similar simplifications can be made for the rest of the conditions. How do I get Mathematica to execute these simplifications on its own?