# Altitudes and Orthocenters

## Jen Curro

In this lesson we will be exploring altitiudes and orthocenters.

Given any triangle ABC

Take the orthocenter (H)

Now connect the vertices of the triangle to the orthocenter forming three smaller triangles inside the larger one.

Now construct the orthocenters of the smaller triangles, AHB, AHC, and BHC. These orthocenters will be labels H', H'', and H'''. Where do you think the orthocenters will be?

The orthocenters appear on the vertices as so

The H' orthocenter is to the triangle, BHC, hence the green dot like the triangle. The H'' orthocenter is to the triangle AHB, hence the red dot like the outlined triangle. And the H''' orthocenter is to the triangle AHC, hence the aqua dot.

Now we are going to construct the circumcirles of the triangles.

To see how this figure was created click here

Next we are going to explore what happens if the vertices were moved to the orthocenter. Click here to examine the movement.

This movement changes the where the orthocenter (H) is located by moving it through the vertex that is being moved and to the outside of the circle. In the same movement the inscribed circle of the vertex being moved moves closer to the original inscribed circle of the triangle ABC. Then when the orthocenter (H) and the vertex being moved are crossed the two circles cross. AT this point, the two points are one and the two circles are one. After this they split and it appears as if vertex being moved takes the path of the orthocenter and H takes the path of the vertex being moved. Eventually the vertex will cross the opposite side of the orginal triangle ABC and form a straight line. If you did not see this in the movement above, click here again and instead of using the movement buttons, manually move one vertex toward the orthocenter and beyond.

We can also explore the four circles of that are formed. To do this click here.

Some other interesting relationships can be explored here.