Converseof the PerpendicularBisectorTheorem:If a pointis equidistantfromthe
endpointsof a segment,then the pointis on the perpendicularbisectorof the segment.Proof. Consider as followswith.We will constructthe midpoint of and showthat is the perpendicularbisectorto.- Constructthe midpointof at point Construct.
- Consider and. Theseare congruenttrianglesby postulateSSS.
- So by CPCTC,we have.
4. Since and form a straight angle and are also congruent, then
. Hence, is on the perpendicularbisectorto.
Noticethat we just provedthe PerpendicularBisectorTheoremand we also provedthe Converseof the
PerpendicularBisectorTheorem.Whenyou provea theoremand its converseyou have provena biconditional
statement.We can statethe PerpendicularBisectorTheoremand its conversein one step:A pointis on the
perpendicularbisectorof a segmentif and only if that pointis equidistantfrom the endpointsof the segment.
We will now use thesetheoremsto provean interestingresultaboutthe perpendicularbisectorsof the sides
of a triangle.
Concurrencyof PerpendicularBisectors:The perpendicularbisectorsof the sidesof
a triangleintersectin a pointthat is equidistantfrom the vertices.Proof. We will use the previoustwo theoremsto establishthe proof.- Consider