This write up is for problems #12 in Assignment 4.

Concurrency of Perpendicular Bisectors of a Triangle

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In this write up we will prove that the three perpendicular bisectors of the sides of a triangle are concurrent.

Given: Triangle ABC with perpendicular bisectors r, s, t

Prove: lines r, s, t are concurrent in a point O and that OA = OB = OC

1. It is given that line r is the perpendicular bisector of segment AB.

2. It is also given that line s is the perpendicular bisector of segment BC.

3. Since segments AB and BC are not parallel, then lines r and s are not parallel. Therefore, lines r and s intersect in a point O.

4. Since a point on a perpendicular bisector is equidistant from the endpoints, then OA=OB and OB=OC.

5. By the transitive property of equality, OA=OC

6. Point O is on the perpendicular bisector ( line t ) of segment AC because a point equidistant from two points is on the perpendicular bisector of the segment determined by those points.

7. Therefore, from statement 3,4, 5, and 6 it has been proved that Point O lies on lines r, s, and t. Likewise, OA = OB = OC.