What is the equation of the angle bisector formed by two intersecting lines?

[V0ODO0CH1LD]

I've seen that if you have two lines: r = Ax + By + C = 0 and r = Dx + Ey + F = 0, you can say the equation of the line that is the angle bisector of r and s is given by: $$\frac{|Ax + By + C|}{\sqrt{A^2+B^2}}=\frac{|Dx + Ey + F|}{\sqrt{D^2+E^2}}.$$
Why is that?

I would think to equate the distances from the angle bisector to each line. Is that what is happening here?

 

[Abhilash H N]

I would think to equate the distances from the angle bisector to each line. Is that what is happening here?

Yes.
Angular bisector of an angle made by two lines is the locus of all points which are equidistant from both the lines. So that the equation of the bisector is given by equating the distances...