Amberly Roberts

Assignment 4: Centroids, Incenters, & Circumcenters

For any triangle, a median can be constructed by creating a segment that runs from a vertex to the midpoint of the opposite side. Since a triangle has three vertices, every triangle has three medians. The intersection of the three medians creates a point of concurrency, called a centroid.

 In triangle ABC, the blue dotted segments are the medians. Their point of concurrency is labeled with the word centroid.   Something to think about... What happens to the location of the centroid as the shape of the triangle changes? Will the centroid always be inside the triangle?       Click here to open a GSP file that allows you to change the shape of the triangle.

For any triangle, an angle bisector can be constructed for each of the three interior angles of the triangle. The intersection of these three bisectors creates a point of concurrency, called an incenter.

 In triangle ABC, the blue dotted rays are the angle bisectors. Their point of concurrency is labeled with the word incenter.   Something to think about... What happens to the location of the incenter as the shape of the triangle changes? Will the incenter always be inside the triangle?       Click here to open a GSP file that allows you to change the shape of the triangle.

For any triangle, a perpendicular bisector is a line that runs through the midpoint of one of the sides of the triangle perpendicular to the side. Since every triangle has three sides, the triangle also has three perpendicular bisectors. The intersection of the perpendicular bisectors creates a point of concurrency, called the circumcenter.

 In triangle ABC, the blue dotted lines are the perpendicular bisectors. Their point of concurrency is labeled with the word circumcenter.   Something to think about... What happens to the location of the circumcenter as the shape of the triangle changes? Will the circumcenter always be inside the triangle?       Click here to open a GSP file that allows you to change the shape of the triangle.

After investigating and exploring these three triangle centers, consider the following question. (This question was posed at a Math I/II training session that I attended earlier this summer.

Here's the problem:

Suppose that a cell phone company has received several complains concerning poor cell phone service from people in three towns in the southern part of Georgia. To keep the clients, the cell phone company hopes to put up another tower to help the service in these three towns. The towns include Fitzgerald, Camilla, and Waycross. Using the information in this investigation, where should the new tower be placed? Justify your answer using mathematical vocabulary.