CENTERS OF A TRIANGLE
Gloria L. Jones
When studying triangles in a standard (high school level) geometry course, students typically study and often construct points of concurrency within a triangle.
The CENTROID (G) is formed at the intersection of the three medians of a triangle:
The ORTHOCENTER (H) is formed at the intersection of the three altitudes of a triangle:
The CIRCUMCENTER (C) is formed at the intersection of the three perpendicular bisectors of a triangle. Notice point (C) is the center of the circum-circle:
Finally the INCENTER (I) is formed at the intersection of the three angle bisectors of a triangle. Notice that point (I) serves as the center for the in-circle as well as the triangle:
In this write-up the author explores these centers of a triangle not as separate and disjoint points of concurrency but, rather as centers often sharing significant geometric/ mathematical relationships.
Using Geometeršs Sketch Pad (GSP) allows much flexibility in constructing multiple diagrams. One can construct a triangle showing all points of concurrency simultaneously and through manipulation of the sketch, relationships are easily seen. The following construction shows all the above points of concurrency in one triangle. Notice the red segment that joins the points (H, G, C) together on one line. The line is known as the Euler line named in honor of Leonhard Euler (Oiler). Also take note that (I) falls outside of the line. The ratio of segments HG:HC is always two thirds of the Euler line and likewise CG:CH is always one third the distance of the Euler line no matter what shape or type of triangle it is:
Considering students have learned the classification of triangles according to sides and angles, they can then start exploring through GSP the behavior of these points of concurrency for various types of triangles.
1. How does the (Euler line) line-up when the triangle is an isosceles?
One can see from the above GSP sketch that all four points of concurrency are collinear on the Euler line.
2. How does the Euler line look in the equilateral triangle?
In the equilateral triangle all points seem to coincide at the same place. It follows then that in the equilateral triangle, the medians, angle bisectors, perpendicular bisectors, and altitudes are all the same.
As shown above here it appears that for an acute triangle all points of concurrency are inside of the triangle.
4. How does the Euler line look for an obtuse triangle?
For the obtuse triangle the circum-center (C) and the orthocenter (H) appear to be outside of the triangle. The Euler line extends beyond the boundary of the triangle.
5. How does the Euler line behave for a right triangle?
For the right triangle above, notice that point (H) lies at the vertex of the right angle and that point (C) lies on the midpoint of the opposite side. The fact that the (H) lies on a vertex and (C) lies on the midpoint of the opposite side, recall, that the segment line (HC) thatšs in this location in particular is a median.