Using problem solving skills, formal geometry, and The Geometer’s Sketchpad, students will model and solve two real-world 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 real-world 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, Intersection, Midpoint, 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.