Assignment eight

Altitudes and Orthocenters

Collaborative Effort by

Lucia Zapata and Claudette Tucker

Problem:

1. Construct any triangle ABC.

2. Construct the Orthocenter H of triangle ABC.

The Orthocenter of a Triangle is constructed by finding the intersection of any two altitudes of a triangle.

3. Construct the Orthocenter of triangle HAB.

Analysis: Note that triangle HBA is an obtuse triangle. It is evident that obtuse triangles have exterior Orthocenters. In contrast, acute triangles have interior Orthocenters. In this case the Orthocenter of HAB is coincident with vertex C.

4. Construct the Orthocenter of triangle HBC.

Analysis: In this case the Orthocenter of HBC is coincident with vertex A.

5. Construct the Orthocenter of triangle HAC.

In this case the Orthocenter of HAC is coincident with vertex B.

6. Construct the Circumcircles of triangles ABC, HBC, HAB, and HAC.

Conjectures:

All four circumcircles are congruent animation

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 an interesting figure.