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Exploring
Centers of a Triangle: Part 1


 
Using problem solving skills,
formal geometry, and The Geometer’s Sketchpad, students will model and
solve two
realworld problems by constructing the incenter and circumcenter of triangles. Students will further explore relationships between
the angle and side bisectors of arbitrary triangles. The author wishes to
acknowledge Dr. Billie F. Risacher for providing the realworld scenarios. Mathematics:
Students will investigate the angle and side bisectors of a
triangle and conjecture the following theorems: ·
The
angle and side bisectors in any triangle are concurrent. ·
The
perpendicular distances from the incenter to each of the three sides of a
triangle are equal. ·
The distances from the
circumcenter to each of the three vertices of a triangle are equal. The activity includes proving the
latter two theorems. Extension
activities include inscribing circles in quadrilaterals and other
polygons. Mathematical
Thinking: Students will be asked to
predict, conjecture and prove the
aforementioned theorems. Technology:
The students
will learn to use several Sketchpad commands: angle
bisector, point
at intersection, midpoint of a
segment, color, length, and distance. The sketch below illustrates the possible location of a pumping station that is an equal distance from each of the oil pipelines.
The sketch below illustrates the possible location of a bus garage that is an equal distance from each of the three high schools.


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