By using Geometer's Sketchpad we can explore various relationships between a triangle and its orthocenter and circumcircle.

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 ABC.

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
the triangle.

Let's look at the four circumcircles generated by our four triangles. Once we have found all four circumcircles, the resulting image looks like:

Why do all of the circumcircles have the same diameter?

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 animations.

Note: you will need Geometer's Sketchpad to view all of the animations described above.

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