9.0 Bisector of angle containing origin

Let equation of two lines be $ax + by + c = 0$ and $a'x + b'y + c' = 0$. Make ${c}$ and ${c'}$ positive.

The bisector of angle containing the origin is given by $$\frac{{{a}x + {b}y + {c}}}{{\sqrt {{a}^2 + {b}^2} }} = \frac{{a'x + b'y + c'}}{{\sqrt {a{'^2} + b{'^2}} }}$$

The bisector of angle does not containing the origin is given by $$\frac{{ax + by + c}}{{\sqrt {{a^2} + {b^2}} }} = - \frac{{a'x + b'y + c'}}{{\sqrt {a{'^2} + b{'^2}} }}$$