Assignment 8

Orthocenters and Circumcircles

Marianne Parsons

By using Geometer's Sketchpad we can explore various relationships between a triangle and its orthocenter and circumcircle.

Orthocenters

First, let's look at an acute triangle ABC and its orthocenter H. The orthocenter is the point defined by the intersection of the altitudes of our triangle.

We now can divide our triangle ABC into three smaller triangles, that all share the vertex H. The three triangles are AHB, BHC, and CHA.

Next, lets find the orthocenter of each of the three smaller triangles. By examining the altitudes of the smaller triangles, we are able to see that the orthocenters of these triangles are actually the vertex points of our larger triangle ABC.

C is the orthocenter of triangle AHB

A is the orthocenter of triangle BHC

B is the orthocenter of triangle CHA

To view the construction of these orthocenters individually, click here.

Circumcircles

We have found the four orthocenters, H, C, A, and B, of our four triangles. Now, let's examine the circumcircles of these triangles. By definition, a circumcircle is a triangle's circumscribed circle that passes through each of the vertices. The center of this circle is called the circumcenter, and is the intersection point of perpendicular bisectors of the triangle.

Let's look at the four circumcircles generated by our four triangles. Once we have found all four circumcircles, the resulting image looks like:

Note points O, d, e, and f, are circumcenters of each of our triangles.

To view the construction of each circumcircle individually, click here. To view the relationship of the circumcenters to each of the four triangles, click here.

What do we observe?

It appears as though all of our circumcircles have the same diameter. Is this always true? Try manipulating the vertices of triangle ABC. Just click on any point A, B, or C, and move the triangle in different directions. What happens to the resulting circumcircles?

Why do all of the circumcircles have the same diameter?

Other Investigations

What happens when triangle ABC is obtuse? View the animation of triangle ABC as it changes from acute to obtuse. What do you see?

Now, view the animation of triangle ABC as it changes from acute to obtuse, but this time notice the different points on the triangles. Watch as the orthocenters, circumcenters, and midpoints move. How do they relate to each other? What do you see? You will need Geometer's Sketchpad to view both of these animations.

What would happen if any vertex on triangle ABC was interchanged with it's orthocenter H? View the animation to watch these points interchange. What do you see?

Note: you will need Geometer's Sketchpad to view all of the animations described above.

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