Assignment 4

Centers of a Triangle

by Rives Poe

In this investigation we are going to explore the relationships of the centers of triangles.

The first center to look at is the CENTROID. (HINT: The centroid is always labeled G. It is labeled G, because there is more than one center so some mathematicians had to get a little creative and it also stands for the center of Gravity of the triangle.)

The centroid of a triangle is the common intersection of the three medians. A median of a triangle is the segment from a vertex to the midpoint of the opposite side. Below is a construction of the centroid.

Click here to see the construction and change the shape and size of the triangle.

The next center we can look at is the ORTHOCENTER. (HINT: The orthocenter is always labeled H.) The ORTHOCENTER of a triangle is the common intersection of the three lines containing the altitudes. (An altitude is a perpendicular segment from a vertex to the line of the opposite side. Below is a construction of the ORTHOCENTER of a triangle.

Click here to explore the orthocenter on GSP with various shaped and sized triangles.

Next we will explore the CIRCUMCENTER of a triangle (always labeled C -finally!) The circumcenter of a triangle is the point in the plane equidistant from the three vertices of the triangle. Since a point equidistant from two points lies on the perpendicular bisector of the segment determined by the two points, C is on the perpendicular bisector of each side of the triangle. Look below to see the orthocenter of a triangle. Notice the circle around the triangle. This is called the circumcircle. It is the circumscribed circle of the triangle, meaning a segment from the circumcenter of the triangle to a vertex creates the radius of the circle.

Click here to use GSP to explore the cirumcenter of a triangle.

Now that we have seen the centers of the triangle, look at them all together. Do you notice any relationship between the points? Why do you think they line up like they do?

The centroid G is in the middle of the circumcenter and orthocenter. Do you think that for any triangle they will form a straight line? Will the line always be inside the triangle? Click here to investigate further on GSP.

Let's take a look at the triangle below. Look at the ratio between segment HG and segment GC.

Notice the triangle formed by A',B', and D'. This triangle is called the medial triangle, because the vertices of the triangle stem from the midpoints of triangle ABD. Therefore, the lengths of triangle A'B'D' sides are half the length of the sides of triangle ABD. The centroid (G) is the same for both triangle. The orthocenter for A'B'D' is actually the circumcenter for triangle ABD. The circumcenter for A'B'D' does not fall on a point on the Euler Line for ABD.