angle bisector as locus

If  ,  then the angle bisector of $\alpha$ is the locus of all such points which are equidistant from both sides of the angle (it is proved by using the AAS and SSA theorems).

The equation of the angle bisectors of all four angles formed by two intersecting lines

$a_{1}x+b_{1}y+c_1=\mathrm{\hspace{0.33em}0},a_{2}x+b_{2}y+c_2=\mathrm{\hspace{0.33em}0}$(1)

is

${\displaystyle \frac{a_{1}x+b_{1}y+c_1}{\sqrt{a_1^2+b_1^2}}}=\pm {\displaystyle \frac{a_{2}x+b_{2}y+c_2}{\sqrt{a_2^2+b_2^2}}},$(2)

which may be written in the form

$x\sin {\alpha }_1-y\cos {\alpha }_1+h_1=\pm (x\sin {\alpha }_2-y\cos {\alpha }_2+h_2)$(3)

after performing the divisions in (2) termwise; the angles ${\alpha }_1$ and ${\alpha }_2$ then the slope angles of the lines.

Note.  The two lines in (2) are perpendicular, since their slopes${\displaystyle \frac{\sin {\alpha }_1\pm \sin {\alpha }_2}{\cos {\alpha }_1\pm \cos {\alpha }_2}}$ are opposite inverses of each other.