Assignment 1

MATH 7200

An Exploration on Centers of Triangles: CENTROiD

by

Samuel Obara

 

The centroid (G) of a triangle is the common intersection of the three medians. A median of a triangle is the segment from a vertex to the mid point of the opposite side.

Construction

Construct any triangle of any size and name it’s vertices as ABC and then construct the medians (DEF) of the three sides of the triangle. Figure 1 shows the centroid of different triangle. It can be noted that G is always inside the triangle whether it is acute or right as shown below. To show more of what I’m illustrating, click animation in the GSP file in this folder.

Most basic property of G is the place of G. Let’s try to see it with some

Calculations. For more demonstration click animation in the gsp file in the centroid folder.

 

 

 

 

As it can be seen from figure 2 that G divides a median into two pieces such that the distance from G to a vertex is twice the distance from G to the midpoint of opposite side of this vertex. In other words, using the same notation in figure 2, GA =2GE, GB = 2GF, and GC = 2GD. This rule does not change with in any triangle, to see this click animation of the gsp file in this folder.

Theorem: prove that the three medians of a triangle are concurrent and that the point of concurrence, the centriod, is two — thirds the distance from each vertex to the opposite side.

Proof:

Let ABC (figure 3) be a triangle and D and F be the midpoint of AB and AC respectively. In triangles ADF and ABC, since angle DAF = angle BAC and AB = 2AD and AC = 2AF, they are symmetric by side — angle — side symmetry theorem. It immediately follows that angle ADF = angle ABC and angle AFD = angle ACB, i.e. DF // BC. (1)

In triangles GFD and GBC, since DF // BC, angle GFD = angle GBC, angle GDF = angle GCB ( from the property of alternative interior angles) angle DGF = angle CGB. So we have

hence