Orthocenter of the Orthocenter

by

Priscilla Alexander

This write-up is for geometry students who are learning about the relationship between the orthocenters of a triangle ,altitudes and circumcenters.

First, to create an orthocenter of a triangle, select a vertex and a side. From this point draw a perpendicular line. Repeat this for all sides. Where the lines intersects is called the orthocenter H. See GSP sketch.

To create a circumcenter take the midpoint of each side and then create a perpendicular line through the midpoint. Where they all intersect is called the circumcenter. The circumcenter C is the center of the circumcircle.

If triangle ABC(the original triangle) is formed and the orthocenter H is created and then triangles HBC, HAB, and HAC. In addition, the circumcircles of these triangles created. See the pictures with the following GSP sketch.

Conjectures

i. A triangle constructed with one vertex at the orthocenter and the remaining two vertices at the vertices of the original triangle, will have its orthocenter at the remaining vertex of the original triangle.

ii. The radii connecting centers of three circumcircles of triangles HBC, HAB, and HBC to the respective vertices of triangle ABC forms a hexagon.

iii. When any of the vertices of the original triangle are moved to the circumcenter of the original triangle a rectangle is formed.

iv. The four circles are congruent.

v. If any of the vertex H4 is moved to C4 with the opposite circumcenter, two congruent triangles are formed.

vi. The area of the hexagon is twice the area of the original triangle.

In conclusion, there are several interesting things about the orthocenters of orthocenters. The above conjectures state what those interesting things are.