We must prove that each pair of triangles formed using this method are congruent. First, we know that the two triangles share a side, which is congruent to itself by identity. In addition, they have another set of sides that, due to the way that the construction took place, must be congruent. Finally, the angle between these sides are congruent using the construction method. Thus, the two pairs of triangles are congruent by SAS. Since they intersect a line, the only way for this to be true is if they intersect at 90 degree angles and if they bisect the lines. Thus, by definition, a perpendicular bisector is created.