Angle bisector/Distance condition/Fact

angle bisector of ${\displaystyle {}v}$ and ${\displaystyle {}w}$ has the same distance to the lines ${\displaystyle {}\mathbb {R} v}$ and to ${\displaystyle {}\mathbb {R} w}$. If a point has the same distance to ${\displaystyle {}\mathbb {R} v}$ and to ${\displaystyle {}\mathbb {R} w}$, then it belongs to the angle bisector of ${\displaystyle {}v}$ and ${\displaystyle {}w}$ or to the angle bisector of

${\displaystyle {}v}$

${\displaystyle {}-w}$