Exploring GSP and the Centers of a Triangle
by
Sharon Sewell
Fall 2001
This will be an exploration of GSP and how it can be used to illustrate triangles and their properties. There are infinite ways to use this piece of software in the classroom. I will take a very simplified route for showing what the program can do.
Let’s take a triangle and explore the centroid, the circumcenter, and the incenter.
The centroid of a triangle is the intersection of the line segments whose endpoints are one vertex of the triangle and the midpoint of the side that is opposite that vertex. These are called the medians. The medians of a triangle intersect in one point that is two thirds of the distance from each vertex to the midpoint of the opposite side.
The centroid would look like this:
The circumcenter of a triangle is the perpendicular bisectors of the sides of a triangle that intersect at one point, which is equidistant from the 3 vertices of the triangle. A note needs to be made that for obtuse triangles, the point of intersection is in the exterior of the triangle.
The circumcenter would look like ...Click here to see it. CIRCUMCENTER
The third triangle center we will look at is the Incenter. This is the three bisectors of the angles of a triangle intersecting at one point that is equidistance from the three sides of the triangle.
The incenter would look like ...Click here to see it. INCENTER
The different centers of the triangles shift and move as the triangle is changed. This is logical because the sizes of the angles will change affecting where the perpendicular lines would be located and the bisectors of the segments. To see this go to the links listed below and play with each of the different centers. Also there is one with all of the centers in one triangle see how each shifts towards and away from one another. Ask yourself at what point do they move outside the triangle if at all.
Here are the GSP scripts for the following triangles. Now is the opportunity to play with these triangles and see how different triangles change.
Triangle with it all (note this triangle has no segments or rays just the points. This is for ease of manipulating and seeing the shifting of the points)
This is a very useful tool for the classroom. So many students are very visual learners and this will physically show them how each point relates to the triangle and each other.