Centers of a Triangle

by Brook Buckelew

1. The CENTROID (G) of a triangle is the common intersection of the three medians. A median of a triangle is the segment from a vertex to the midpoint of the opposite side.

2. The ORTHOCENTER (H) of a triangle is the common intersection of the three lines containing the altitudes. An altitude is a perpendicular segment from a vertex to the line of the opposite side. (Note: the foot of the perpendicular may be on the extension of the side of the triangle.) It should be clear that H does not have to be on the segments that are the altitudes. Rather, H lies on the lines extended along the altitudes.

Here is the relationship between the centroid(G) and orthocenter(H) for this particular triangle.

3. The CIRCUMCENTER (C) of a triangle is the point in the plane equidistant from the three vertices of the triangle. Since a point equidistant from two points lies on the perpendicular bisector of the segment determined by the two points, C is on the perpendicular bisector of each side of the triangle. Note: C may be outside of the triangle.

This is the construction for point C.

Look at what happens when the point C is outside of the triangle.

Notice the relationship between the points H, G, C.

4. The INCENTER (I) of a triangle is the point on the interior of the triangle that is equidistant from the three sides. Since a point interior to an angle that is equidistant from the two sides of the angle lies on the angle bisector, then I must be on the angle bisector of each angle of the triangle.

More examples of the incenter:

5. 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.

Here is the construction of Points G, H, C, and I.

Let's take away our construction lines so that we can see it better.

Notice what happens when one of the angles of our triangle is 90 degrees. Points C, G, and H are on the same line.

What kind of triangle do we have when all four points are meeting at the exact same position?

That's right! It's an equilateral triangle.