Write-Up #4
Centers of a Triangle
By Jaepil Han
1. The CENTROID (G) of a triangle is the point of concurrency of the three medians. A median of a triangle is the segment from a vertex to the midpoint of the opposite side. Use Geometer's Sketchpad (GSP) to Construct the centroid and explore its location for various shapes of triangles. Some ideas for write-ups about Centroids:
-- The medians divide the triangle into six small triangles. Show that these triangles all have the same area.
Let G be the centroid of a triangle ABC.
Since the centroid G is the point of concurrency of the three medians, we may easily find the three medians, M, N, and S, of triangle ABC.
Consider the bottom triangle GBC.
The triangle GBC consists of the triangle GBM and GCM. Since the two triangles have the same altitude from G to BC and the point M bisects the segment BC, the areas of triangle GBM and triangle GCM are the same.
With similar reason, the triangle ABM and triangle ACM have the same area.
Now, consider some parts of these triangles. The triangle ABM consists of the triangle ABG and triangle GBM. Also, the triangle ACM consists of the triangle ACG and triangle GCM.
Since the areas of triangle GBM and triangle GCM are the same, the area of triangle ABG is the same as the area of triangle ACG.
Since the point S bisects the base of the triangle GAB, the area of triangle GAS is the same as the area of triangle GBS. Similarly, the area of triangle GAN is the same as the area of triangle GCN.
So, the area of triangle GAS is the half of the area of triangle GAB and the area of triangle GAN is the half of the area of triangle GCA. Since, the areas of triangle GAB and triangle GAC are the same, the area of triangle GAS is the same as the area of triangle GAN.
Similarly, we can easily show that the areas of triangle GBS and triangle GBM are the same and the areas of triangle GCM and triangle GCN are the same.
Thus, all the small triangles have the same area.
Therefore, the area of the small triangle is 1/6 of the area of triangle ABC.
