Altitudes and Orthocenters
by
Rob Walsh
We seek to find relationships between the orthocenter of a given triangle and subsequent orthocenters of triangles constructed within the given triangle. Let's begin with a GSP construction of the following:
I created a script tool based on the following steps that will reproduce the above construction:
1. Construct any triangle ABC.
2. Construct the Orthocenter H of triangle ABC.
3. Construct the Orthocenter of triangle HBC.
4. Construct the Orthocenter of triangle HAB.
5. Construct the Orthocenter of triangle HAC.
6. Construct the Circumcircles of triangles ABC, HBC, HAB, and HAC.
Let's consider a few observations regarding the relationship between the triangle vertices and the various orthocenters in our construction. Primarily, during the construction we notice that the orthocenters of each of the triangles having a vertex point H are in fact the vertices of our initial triangle ABC. What follows from this is that if orthocenter H exchanges positions with any vertex of triangle ABC, then the positions of the triangle's circumcircles switch. In effect, orthocenter H of triangle ABC becomes vertex H for triangle ABC and simultaneously becomes the orthocenter for one of the triangles HBC, ABH or ACH (depending on which vertex it changes places with.)
In doing this construction, several common points surfaced amongst the circles. Namely, the midpoints of the segments forming each of the four triangles were observed to have a connection. Specifically, our earlier construction of the nine point circle (see assignment 5) used all six of the points in the construction of a circle. Let's find out where it may get the other three points...
First, we look at the nine point circle for triangle ABC. Since we have already noted that triangles HBC, ABH and ACH are essentially created by exchanging positions of triangle ABC's orthocenter and one of its vertices, we will generalize later. Our GSP file has a tool to quickly recreate the construction. Upon doing so, we see six of the nine points on the nine point circle of triangle ABC. We know this to be the case, because these points represent the midpoints of the triangle's three sides and the midpoints of the segments from the triangle's orthocenter to each vertex. The latter three points were used in design of the mid-segment triangle tool while the former three are standard in locating the centroid. However, we still need three more points; namely, we are missing the feet of the altitudes of triangle ABC. Of course, these were used in the construction of the orthocenter itself. So, we have the nine points we need:
Our nine point circle!
Feel free to walk slowly and step-by-step through the GSP file and do some more exploration on your own!