In this section, we are constructing and investigating the centroid of various triangles. The centroid of a triange is defined to be the common intersection of the three medians. The Geometer's Sketchpad is a nice software package to use when trying to construct the centroid. Once the triangle has been constructed and labeled, select one of the sides and use the CONSTRUCT feature to create the midpoint of the selected side. Next, construct the median by selecting the midpoint (which should already be selected) and the vertex opposite the selected side. Again, use the CONSTRUCT feature to create the segment with the two highlighted points as endpoints. Repeat this procedure for the other two sides and vertices. The picture below is an example of a centroid.

In the diagram above, the point H is the centroid of the triangle.

Next, let's examine the location of the centroid in an obtuse triangle. Again, the constructions are performed the same way as with the acute triangle.

In this diagram, the centroid is labeled K. In the two triangles above,
the centroid was located in the interior of the triangle.

Let's now investigate what happens with a right triangle.

Here, the centroid has been labeled Z. We are assured that this is a
right triangle because the measure of one of the angles is 90 degrees. As
before, the centroid is located in the interior of the triangle.

It might also be interesting to investigate whether or not there are any relationships found within the segments that determine the centroid of a triangle. Take some arbitrary triangle, say triangle ABC with centroid J, and divide each median into two parts. Next, measure each segment.

Now, consider the following ratios:

Notice that the ratio is 2:1.

After the three medians are constructed in the original triangle, we notice that the original has been subdivided into six triangles. Hence, we may calculate the area of each region.

It follows that each triangle has the same area.