Triangle/Perpendicular bisector/Intersecting point/Fact/Proof

The perpendicular bisector for the line segment between ${\displaystyle {}A}$ and ${\displaystyle {}B}$ consists, according to fact, exactly of those points of the plane that have to these points the same distance. Therefore, the intersecting point ${\displaystyle {}P}$ of the perpendicular bisector of ${\displaystyle {}A}$ and ${\displaystyle {}B}$ and the perpendicular bisector of ${\displaystyle {}A}$ and ${\displaystyle {}C}$ has the same distance to all three vertices. This gives the supplement, and also, that all three perpendicular bisectors meet in this point.