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If we have two lines given by the equations:

$$ax+by+c=0$$

$$px+qy+r=0$$

We know that the two angle bisectors are represented by the equations

$$\frac{ax+by+c}{\sqrt{a^2+b^2}}=\pm\frac{px+qy+r}{\sqrt{p^2+q^2}}$$

Is there any simple way to distinguish between the two angle bisectors? That is, can we tell which sign will give the acute angle bisector and which the obtuse? I tried using the fact that the angle between the acute angle bisector and one of the lines will be less than 45 degrees,but the formula for that is too hairy.

I remember reading somewhere that if $c, r$ are of the same sign and $ap +bq$ is negative then the positive sign gives the acute cangle bisector. I tried it out for some values in a graphing programme and it does seem to work. But I was not able to come up with a proof.

Gerard
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  • they ought to be orthogonal to each other. depends on which bisector you want. if you need the within the acute angle, pick a point on one line and drop the perpendiculars from it see which one is shorter. – abel Jan 19 '15 at 14:46
  • @abel: Yes, we can do that. However, that will not be a simple computation. The formula I mentioned in my question is much more succint. – Gerard Jan 19 '15 at 15:27
  • what you have is the distance from a point on one the bisectors to the two lines. what you need to discriminate between the two bisectors, only thing i know is the i suggested in the previous comment. – abel Jan 19 '15 at 17:58

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