The proof is below.
Work Step by Step
We must prove that no matter what point on the perpendicular bisector is being considered, the two triangles formed by the bisector are congruent. For starters, 90 degree angles are congruent to each other, so the two right angles formed by the bisector are congruent. In addition, the side that the two triangles share is congruent to itself by identity. Finally, since it is a bisector, the two lines formed by the bisector are congruent. Thus, the two triangles are congruent by SAS, meaning that any point on the line will always be equidistant from either point and thus making this method valid.