EMAT 6680 Assignment 4
Centers of a Triangle
This Assignment has a set of
activities to help become familiar with GSP and to review some basic geometry.
After examining the activities in the assignment, pick some topic for a brief
write-up. The write-up could be one of the proofs, but does not have to be. It
could be some exploration you would try with students. Or you might take one of
the topics (e.g. medians) and explore some of the standard geometry for the
topic.
Theorem: The perpendicular
bisectors of the sides of a triangle are concurrent at a point equidistant from
the vertices
12. Prove that the three perpendicular bisectors of
the sides of a triangle are concurrent.
Given: Lines l, m, and n are
perpendicular bisectors of the sides of triangle ABC.
Prove: That lines l, m, and n
intersect at one common point.
1. We are given that line l is the perpendicular bisector of
segment AB.
2. We are also
given that line m is the perpendicular bisector of segment AC.
3. Since segments
AB and AC intersect at point A of triangle ABC, they are not parallel to each
other by definition of parallel.
4. Therefore, their
perpendicular bisectors l and m must also intersect at some
point. We will call that point of intersection point D.
5. Since l is the perpendicular bisector of segment AB, DA = DB
by the perpendicular bisector theorem.
6. Since m is the perpendicular bisector of segment AC, DA =DC
by the perpendicular bisector theorem.
7. Thus DA=DB=DC, by the transitive property of equality.
8. Since
DB=DC, point D is on line n by the converse of the perpendicular
bisector theorem.
9. Therefore lines l, m,
and n intersect at one common point X.
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