Altitudes and Orthocenters

By

Brooke Norman

The altitude is also known as the height of a
triangle. It is a line perpendicular to
a side of the triangle that goes through the opposite vertex of a triangle.

There are three altitudes in every triangle. The intersection of these three is called the
orthocenter. It is usually labeled,
H.

As you can see, the triangle is now broken up into three
smaller triangles. The are triangles
AHB, AHC, and BHC.

We will now find the orthocenters of each of these
triangles.

As you notice, the orthocenters of the three inner
triangles lie on the vertices of the original triangle.

The
orthocenter of triangle AHB is C

AHC is B

BHC is A

We will
not construct the circumcircles of all four triangles. It should look like this:

If you connect the origins of the three circumcircles to
their nearest vertices of the original triangle ABC and to the orthocenter, H,
you form a cube!

You may play around with this construction by clicking here. Try moving the
different points around and see what happens.

Here are some interesting things
to take note of:

Each of the four circumcircles have the same radius.

The orthocenters of the three inner triangles in the original triangle lie on the vertices of the original triangle.

The
lines going from the orthocenter H of the original triangle from the center of
each circumcircle, and the two closest vertices of the original triangle,
form a cube.

The lines used to construct the orthocenter of the original triangle ABC bisect the overlapping areas of the outer circumcircles.

Return to Brooke’s EMAT 6680
Homepage.