# Altitudes and
Orthocenters

# by

# Mindy Swain

__Assignment 8__

1. Construct any triangle ABC.
2. Construct the Orthocenter H of triangle ABC.
3. Construct the Orthocenter of triangle HBC. (it is A)
4. Construct the Orthocenter of triangle HAB. (it is C)
5. Construct the Orthocenter of triangle HAC. (it is B)

Proof that Orthocenter of triangle HBC is A:

By construction AH is perpendicular to BC,
AB is perpendicular to CH,
and AC is perpendicular to BH,
where BC, CH, and
BH are the three sides
of triangle HBC.

So, the orthocenter of HBC
must lie on AH, AB, and
AC.

Therefore, the intersection
of these three lines is the orthocenter, **A**.

A similar proof follows for
HAB and HAC.

6. Construct the Circumcircles of triangles ABC, HBC, HAB, and HAC.

7. Construct the Nine Point Circles of triangles ABC, HBC, HAB, and HAC.

__Conjecture__: The
Nine Point Circles of triangles ABC, HBC, HAB, and HAC are the
same.
__Proof__:

By definition, the nine point
circle bisects any line from the orthocenter to a point on the
circumcircle. [from Math World]

Also, the radius of the nine
point circle is one half of the circumradius of the reference
triangle. [from Math World]

Using GSP we can utilize the
script tool from my assignment 5 to see the nine point circles
of these four triangles. They seem to all be the same.

By dilating each of the circumcircles
with its corresponding orthocenter by a scale factor of 1/2,
you will notice that the circle that is created lies on the nine
point circle.

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