TITLE: How do I find the slope of an angle bisector, given the equations of the two lines that form the angle?
QUESTION [7 upvotes]: The equation for the first line is $y = \frac{1}{2}x - 2$, and the equation for the second line is $y = 2x + 1$. They intersect at $(-2, -3)$.
Someone told me I can just average the slopes of the two lines to find the slope of the bisector, but I'm not sure if it's right.

REPLY [3 votes]: To be on right track, let $Ax + By + C =0, ax + by + c =0 $ be the equations of the given straight lines.The angular bisector is a straight line locus so that length of perpendiculars dropped from it onto the given lines are equal. The polar form of straight line is useful here.
$$
\frac{Ax+By+C}{\sqrt{A^2+B^2}} = \pm \frac{ax+by+c}{\sqrt{a^2+b^2}}.
$$
The $+$ sign is taken when the arms contain the origin for internal bisector, $-$ sign for the external bisector perpendicular to it. You can now find its slope.