By using Geometer's Sketchpad we can explore
various relationships between a triangle and its orthocenter and
First, let's look at an acute triangle ABC
and its orthocenter H. The orthocenter is the point defined by
the intersection of the altitudes of our triangle.
Next, lets find the orthocenter of each of
the three smaller triangles. By examining the altitudes of the
smaller triangles, we are able to see that the orthocenters of
these triangles are actually the vertex points of our larger triangle
We have found the four orthocenters, H, C,
A, and B, of our four triangles. Now, let's examine the circumcircles
of these triangles. By definition, a circumcircle is a
triangle's circumscribed circle that passes through each of the
vertices. The center of this circle is called the circumcenter,
and is the intersection point of perpendicular bisectors of
Let's look at the four circumcircles generated
by our four triangles. Once we have found all four circumcircles,
the resulting image looks like:
appears as though all of our circumcircles have the same diameter.
Is this always true? Try manipulating
the vertices of triangle ABC. Just click on any point A, B,
or C, and move the triangle in different directions. What happens
to the resulting circumcircles?
Why do all of the circumcircles have the same
happens when triangle ABC is obtuse? View the animation of triangle
ABC as it changes from acute
to obtuse. What do you see?
Now, view the animation of triangle ABC as
it changes from acute to obtuse, but this time notice the different points
on the triangles. Watch as the orthocenters, circumcenters, and
midpoints move. How do they relate to each other? What do you
see? You will need Geometer's Sketchpad to view both of these
would happen if any vertex on triangle ABC was interchanged with
it's orthocenter H? View the animation to watch these points
interchange. What do you see?
Note: you will need Geometer's Sketchpad to
view all of the animations described above.