*Nine-Point
Circle*

* *

*By Courtney
Cody*

In this assignment, we will look
at the construction of the nine-point circle and locate the center, N. In order to understand what we are
going to construct, we must begin with a good definition of *nine-point
circle* before we can begin our
construction.

**By definitionÉ**

á
The *nine-point circle* for any triangle passes through the three midpoints
of the sides, the three feet of the altitudes, and the three midpoints of the
segments from the respective vertices to the orthocenter.

WeÕll begin our construction
with an arbitrary triangle ABC

Since by definition the nine
point circle is suppose to pass through the three midpoints of the sides of a
given triangle, we will use the capabilities of GeometerÕs Sketchpad (GSP) to
construct the midpoints of segments AB, BC, and CA and call them D, E, and F,
respectively.

Next, the definition requires
that we construct the three feet of the altitudes. By definition, an *altitude* is a line that extends from one vertex of a triangle
perpendicular to the opposite side.
Using GSP, we will construct the three altitudes of our triangle and
call the points where each of the altitudes intersects each side G, I, and
J. Additionally, the intersection
of the three altitudes is called the *orthocenter* and is labeled point H, by convention.

If we recall, the third
component of the definition of *nine-point circle* asks that we find the three midpoints of the segments
from the respective vertices to the orthocenter. In order to do this, we first need to find these three
segments, which are pictured on the diagram in **dark
green**. The midpoint of
each of these segments is also pictured and we call them K, L, and M.

At this time, we have located
the nine points of the *nine-point circle*. Now, we need to find the
center, *N*. One way to find the center of the nine
point circle is to first find the circumcenter O, which is the intersection of
the three perpendicular bisectors of triangle ABC. Next, we find the *Euler line*, which is the line passing through the *orthocenter* *H* and the
*circumcenter* *O* of our triangle ABC. This line is pictured in dark red. The midpoint of the *Euler line* is the *center of the nine-point circle, *called *N*,
as desired

Lastly, we arrive at the
complete illustration of our *nine-point circle*É.

Suppose our original triangle
had a different shape. Would this
change the appearance of the *nine-point circle*? The
triangle we assumed at the beginning of our construction was an acute triangle,
meaning all three angles of the triangle were acute. When this is the case, three of the nine points lie inside
the triangle while the other six lie on the triangle. Specifically, points L,
K, and M are inside the triangle.
Now, suppose we change the triangle such that triangle ABC becomes
obtuse. Then we notice that the
same three of the nine points that were previously interior to the triangle
move to the exterior of the triangle.

When triangle ABC is a right
triangle, all nine points of the *nine-point* *circle*
lie on the triangle, which is illustrated below:

Click here
to see for yourself how the *nine-point circle* change as triangle ABC is moved around.