Altitudes
and Orthocenters

(Assignment 8)

**by**

J. Matt Tumlin, Cara Haskins, and Robin Kirkham

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

Now, construct
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 whether it is acute, right, or obtuse?

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

To use a
script tool to find out, CLICK HERE!

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?

To use a script tool to
move the points, CLICK HERE!

In conclusion
some conjectures are:

á The radii of
each 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 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.