**Mathematics
Education**

**EMAT 6680,
Professor Wilson**

**
Exploration 8, Altitudes and Orthocenters by Ursula Kirk**

a.
Construct
any triangle ABC

b.
Construct
the orthocenter of triangle ABC

c.
Construct
the orthocenter of triangle HBC

d.
Construct
the orthocenter of triangle HAB

e.
Construct
the orthocenter of triangle HAC

f.
Construct
the circumcircles of triangles ABC, HBC, HAB and HAC.

g.
What
would happen if any vertex of the triangle ABC was moved to where the
orthocenter H is located?

When the
orthocenter coincides with any vertex of triangle ABC, all the triangles HBC,
HAC and HAC coincide with triangle ABC as it can be observed on the picture
below. In addition, the circumcircles of triangle ABC and HBC coincide. Also
the circumcircles for HAC and HAB become tangent at point A and H. Therefore A
= H and orthocenterABC=orthocenterHAB

**The nine point
circle**

It is a circle
that passes through nine critical points of a triangle. These points are

1.
The
3 midpoints of the side of the triangle

2.
The
3 feet of the altitudes of the triangle

3.
The
midpoint of the line segment that joints each vertex to the orthocenter.

For triangle
ABC, the nine point circle goes trough points L, M and N which are the
midpoints of the triangle. Also goes through X, Y and Z which are the midpoints of the segments AH, BH, CH. Finally it also goes
through the points D, E and F which are the feet of the altitudes of triangle
ABC. Using O as the center of my nine point circle, I can construct a circle
that passes through all of these critical points.

Another interesting construction in the picture above is the
Euler segment. By constructing the orthocenter, centroid, and circumcenter of
triangle ABC, I find a segment where the orthocenter, the centroid and the
circumcenter are collinear. The incenter is not on the Euler line except when
the triangle is isosceles.