Perpendicular
Bisectors
Assignment 4 Ð Diana
May
The perpendicular bisectors of a triangle ABC are the lines that pass through the midpoint of each side which are also perpendicular to that given side.
The circumcenter is defined as the intersection of the three perpendicular bisectors of a triangle ABC, but how do we know that these three perpendicular bisectors intersect at a single point?
Claim: The perpendicular bisectors of a triangle ABC are concurrent.
Proof:
*For any two points A and B, the set of points equidistant from A and B is the perpendicular bisector of AB.
Label the perpendicular
bisectors of AB and BC as m and n respectively. Also, let the intersection of m and n be P.
Then AP = BP
since P is on m and BP = CP since P is on n (from *).
Thus AP = BP = CP. But AP = CP implies that P is on the perpendicular bisector of AC (from *).
This shows that all 3 perpendicular bisectors pass through P, so they are concurrent at P.
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