CENTERS OF A TRIANGLE
Sharren M. Thomas
In a standard Geometry course, students typically study and often construct points of concurrency for the Centroid (G), orthocenter (H), Circumcenter (C), and finally the Incenter (I).
In this write-up, I will explore with my students the various points G, H, C, and I and their mathematical relationships in several different triangles. Additionally, we will explore the Euler line which is the segment joining C, G, and H.
This comparison will most likely take place after sections in which students have already learned to classify a triangle according to its sides and angles. GSP (Geometry Sketchpad) will be used to explore the behavior of these concurrent points (G, H, C, & I) of a equilateral, right, acute, and obtuse triangles.
See below, a triangle which shows the centroid, orthocenter, circumcenter, and the incenter for the same equilateral triangle. Notice that G, H, C, and I coincide.
In a right triangle H, the orthocenter corresponds to the 90º vertex angle. C, the Circumcenter is located at the midpoint of the hypotenuse, G and I are located within the triangle. See below.
Now we will explore the location of these points in an acute (non-equilateral) triangle.
Notice that all concurrent points lie within the triangle, and the Centroid and the Circumcenter overlap for this case, but not for all cases of acute triangles.
Last investigation explores the concurrent points in an obtuse triangle.
Notice that both the Circumcenter, C, and the Orthocenter, H, are located outside the triangle.
Recall, in a right triangle H lies on a vertex and C lies on the midpoint of the opposite side; thus, segment HC would be considered a median of the right triangle. Thus G should be 2/3 the distance from H to C as the centroid. This a particular properties that holds true. Is G always 2/3 the distance from H to C on the Euler line regardless of whether it is a right triangle or not.
Click here to manipulate the sketch by moving a vertex and analyze the ratio of HG to HC.