Exploring Altitudes and Orthocenters

by

Doug Westmoreland

This is the write-up for assignment #8

The first thing that we want to do in the exploration is to construct a triangle ABC and its orthocenter H.

Next, we now want to construct the orthocenters of triangle HBC, triangle HAB, and triangle HAC.

First thing that you should notice is that the orthocenters H1, H2, and H3 are located at the vertices of the original triangle ABC. Click here for a gsp sketch that you can drag around and observe the location of the orthocenters.

The next thing that we are gonna do is to construct the circumcircles of triangles ABC, HBC, HAB, and HAC.

The dotted red circle is the circumcircle to triangle ABC. The dotted green circle is the circumcircle to triangle HBC. The dotted black circle is the circumcircle to triangle HAB. The dotted blue circle is the circumcircle to triangle HAC.

For a gsp sketch that you can drag around and observe the behavior of the circumcircles of the drawing above click here.

Now, lets take a look at the 9-point circles for triangles ABC, HBC, HAC, and HAB. In order to see the behavior of these circles we must first take a point D and construct the 9-point circles for the triangles ABC, DBC, DAC, and DAB. In the figure below this construction is given. All of the dotted circles are the 9-point circles for triangles ABC, DBC, DAC, and DAB. The point H is the orthocenter to triangle ABC.

Now, how do you think that the 9-point circles will behave as we let point D get closer and closer to point H until point D lies directly on top of point H? Click here for a gsp sketch that will allow you to drag point D onto point H.

In the drawing below, a triangle ABC was constructed along with its incircle, its three excircles, and its nine-point circle.

Now, if we were to drag the vertices of triangle ABC around, what do you think the behavior of the incircle, 9-point circle, and the three excircles will be?

Do you think that the 9-point circle will always be tangent to the excircles? For a dynamic presentation of the above figure using GSP click here.

What happens to the excircles when the incircle and the 9-point circle are the same? Conjectures?

Lets now take a look at the orthic triangle and the triangle formed by the points where the extended altitudes meet the circumcircle.

In the figure below, we see the triangle ABC (blue) and its orthic triangle GHI (grey). The altitudes of triangle ABC were extended to intersect its circumcircle (points D, E, and F).

Click here for a GSP sketch that you can drag the points A, B, or C around and see how the orthic triangle relates to triangle DEF.

Return