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I have the following Mathematica result under the assumption that {ax, bx, px, ay, by, py} \[Element] Reals:

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:

enter image description here

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?

This is the question I'd like to ask, but I am not sure how to ask it well to be honest. For instance: is there another question that is very similar to mine (likely, but I can't tell from what I see). More importantly, what should I title this question? Here's what I think:

  • How to get Mathematica to make further simplifications to logical statement?

It's vague though, and I have had a bit of a history with poor questions recently that is making me very self-conscious.

  • 2
    If your self-consciousness is due to the site's automated warnings, "you may be banned" &c., please do not take it too seriously. We do not agree with those warnings being given automatically and nobody is going to ban you because of it. It's obvious from the above that you do know how to ask a good question, but worth noting that beauty is as much in the eye of the beholder as the possessor. So, I suggest that you make your question as incisive as possible from your own point of view, and just ask it. It may or may not get a good response, but this is not necessarily your responsibility. – Oleksandr R. Sep 23 '15 at 23:27
  • 3
    I will add though that you should make sure you do not have any syntax errors or misleading statements in the question, and that it is not unnecessarily verbose. A shorter and more direct question is normally more compelling, all else being equal. If there are subsidiary problems, it distracts from your main query and can lead to people being confused about what you really need, which may result in the question being closed. Sometimes closure is necessary, but even if it happens, you should be aware that this is not intended as a moral judgment on you. A question can always be reopened. – Oleksandr R. Sep 23 '15 at 23:32

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