Assignment 8: Orthocenters

 

Jason P. Pickhardt


Problem:

 

Complete the following constructions:

 

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.

 

What would happen if any vertex of the triangle ABC was move to where the orthocenter H is located? Where would H then be located? 

 

Exploration:

 

The constructions required were made in Geometerís Sketch Pad here. The constructions of the orthocenters of triangles HBC, HAB and HAC yielded some interesting observations. It turns out in fact that the orthocenters of these triangles are the vertices of the original triangle ABC.

 

The orthocenter of HBC is vertex A.

 

The orthocenter of HAB is vertex C.

 

The orthocenter of HAC is vertex B.

 

While constructing the circumcircles of each triangle we have to construct the midpoints of all the segments of the triangles. If you think about how these midpoints are all related it is possible to realize that these are six of the nine points on the nine point circle. The midpoints of segments AH, BH and CH are the midpoints of the segments from each vertex to the orthocenter, and we also have the three midpoints of triangle ABC. It remains to construct the three points on the feet of the altitudes of triangle ABC. This is very simple since the altitudes of triangle ABC were drawn in order to construct the orthocenter of this triangle. The nine point circle can be seen in yellow is the previous GSP file.

 

Note: the color of each circumcircle corresponds to its triangular interior except for triangle ABC whose circumcircle is dark blue.

 

Conclusion:

 

After the constructions are finished it remains to explore what happens when any of the vertices are relocated to orthocenter H. Using the given GSP file, drag any of the vertices to the approximate location of point H. At first glance one can conjecture that the measure of the angle at this new vertex is 90 degrees. We can confirm this by noticing the measurements given within the file. Of course these are approximate measurements since we cannot be sure that the given vertex is at exactly point H. This occurs for any of the three vertices. We can also see that the new orthocenter of the triangle is located still at point H or the vertex that is 90 degrees. Therefore we can conjecture that the orthocenter of a right triangle is always at the vertex of the right angle.

 

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