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

**Concurrency of Perpendicular Bisectors
of a Triangle**
by
**Brad Simmons**

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

To review the definition of **perpendicular
bisector of a segment** and **concurrent** please click
here.

**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.

For a dynamic sketch that can be manipulated
please click
here.

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