Triangle/Perpendicular bisector/Intersecting point/Fact/Proof

Proof

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

To fact