Assignment 8:

Altitudes and Orthocenters

by Kristina Dunbar, UGA

Part One

The altitude of a triangle is a line perpendicular to a side of the triangle that goes through the opposite vertex of a triangle.  It is also called the height.

Every triangle has three altitudes, and the altitudes of a triangle intersect at the orthocenter.  The orthocenter is usually denoted H.

Assignment 4 had an in-depth discussion of triangle centers, including the orthocenter.  To review, click here.

As we can see, the orthocenter breaks the triangle ABC up into three new triangles, ABH, BCH, and CAH.  Let's look at those.

Now, let's find the orthocenters of each of these new triangles.

So, what did we discover?  The orthocenters of the three new triangles lie on the vertices of the original triangle!

• The orthocenter of ABH is at point C.
• The orthocenter of BCH is at point A.
• The orthocenter of CAH is at point B.

What if we changed the shape of the original triangle?  Would we still get the same result?

The above two examples show that the orthocenters of the three new triangles remain on the outer vertices of the larger triangle.  Even in the case of an obtuse triangle ABC, where the orthocenter is outside of the original triangle, we see the same result.

If you would like to explore this for yourself, click here to download the GSP construction.

Part Two

Let's construct the circumcircles of our four triangles above.  What will they look like?

Click here for a GSP version of this construction.

Part Three

An interesting thing happens when you connect the origins of your three circumcircles to their nearest vertices of the original triangle ABC and to the orthocenter H of triangle ABC.  You form a cube!

Click here to see this construction using GSP.  The cube is not always regular, which you will see if you click the animate button.

What did we notice from the above constructions?

• All four circumcircles have the same radius.

• The lines used to construct the orthocenter of the original triangle ABC bisect the overlapping areas of the outer circumcircles.

• The orthocenters of the three triangles formed by creating the orthocenter of the original triangle lie on the vertices of the original triangle.

• The lines from the center of each circumcircle going to the orthocenter H of the original triangle and the two closest  vertices of the original triangle form a cube.