Prove that the three Perpendicular Bisectors of the sides of a triangle are concurrent.

(Sub-Proof)

by

Summer Brown


 
 




We wish to prove that segment AC* is congruent to segment BC*.  We will use this congruence later in the main proof.
 


Hence, by the Side-Angle-Side (SAS) congruence criterion, Triangle ADC* is congruent to Triangle CDC*.  Since corresponding parts of congruent triangles are congruent, AC* is congruent to CC*.

Similarly:


Hence, by the Side-Angle-Side (SAS) congruence criterion, Triangle BFC* is congruent to Triangle CFC*.  Since corresponding parts of congruent triangles are congruent, BC* is congruent to CC*.

Since AC* is congruent to CC* and BC* is congruent to CC*, AC* is congruent to BC* by transitivity.  This makes sense since AC*, BC*, and CC* are all radii of the circumcircle.



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