Altitudes and Orthocenters

(Assignment 8)

by

Robin Kirkham,

J. Matt Tumlin, and Cara Haskins,

An orthocenter (H) of a triangle is the point where the lines containing its altitudes are concurrent.

Now, let us look at the construction of triangles between the orthocenter and 2 of the vertices of the triangle.

As you can see, the orthocenters of the three triangles are the vertices of the original triangle.

Does this relationship hold true for triangle ABC given that a triangle may be acute, right, or obtuse?

What would happen if one of the vertices of triangle ABC was moved to where the orthocenter (H) is located?

Now construct the Circumcircles for DABC, DHBC, DHAC, and DHAB.

What can you say about the radii of the circumcircles?  Are they congruent?  Are you sure?

In conclusion some conjectures are:

á      The radii of the circumcircle is exactly the same.

á      All circumcircles pass through the orthocenter (H) of the original triangle ABC.

á      A cube is formed by connecting the centers of the circumcircles with the orthocenter (H).  To use a script tool, CLICK HERE!

á      The area of the overlap of the circumcenters at each vertex is bisected by the altitudes of the original triangle.

á      A second area of the overlap of the circumcircles is bisected by the sides of the original triangles.

á      We know that the perpendicular bisector that intersects each side of the original triangle and travels through the orthocenter also intersects one of the centers of the outside circumcircles.