### Assignment 8:

# Altitudes and Orthocenters

#### By

## 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.

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