Consider the triangle [math]\triangle ABC[/math]. Let the perpendicular bisectors of the sides [math]\overline{AB}[/math] and [math]\overline{BC}[/math] intersect at point O. [br][br]Because [math]\overline{OA}\cong\overline{OB}[/math] and [math]\overline{OB}\cong\overline{OC}[/math], we have [math]\overline{OA}\cong\overline{OC}[/math]. Now consider the point F that bisects the segment [math]\overline{AC}[/math]. Because it is a bisector, we know that [math]\overline{AF}\cong\overline{CF}[/math]. We also know that [math]\overline{OF}[/math] is congruent to itself. Thus, by the SSS criterion for congruence, [math]\triangle AOF\cong\triangle COF[/math].