EMT 668 - ASSIGNMENT 4, #5
Centers of a Triangle

by

Kimberly N. Bennekin


Use GSP to construct G, H, C and I for the same triangle. What relationships can you find among G, H, C, and I or subsets of them? Explore for many shapes of triangles.

The point G is the CENTROID of a triangle. It is the common point of the three medians.

The point H is the ORTHOCENTER of a triangle. It is the common intersection of the three lines containing the altitudes. An altitude is a perpendicular from a vertex to the line of the opposite side.

The point C is the CIRCUMCENTER of a triangle. It is the point in the plane equidistant from the three verticies of the triangle.

The point I is the INCENTER of a triangle. It is the point on the interior of the triangle that is equidistant from the three sides.

RELATIONSHIPS

We can examine the behavior of G, H, C and I by examining their position as the triangle transforms shape. Consider the following triangles.

i) For an equilateral triangle, G, H, C and I are all the same point.

ii) For a right triangle, the orthocenter H occurs at the vertex of the triangle where we have
the 90 degree angle. The circumcenter C occurs on the hypotenuse of the right triangle.

iii) For various other triangles, H ranges from being inside the triangle, on a vertex of a
triangle or outside of the triangle. The point C ranges from being inside the triangle,
on one side of the triangle, or outside of the triangle. The points G and I are restricted
to inside the triangle. (See figures below.)



iv) We can see the how these points line up by creating a line segment with two of the points. G, H and C always line up. G is always two thirds the distance from H to C. With isoceles triangles, I will line up as well. Consider the figures below.




Return to Class Page