Thomas Earl Ricks

Mathematics Education

Assignment # 4

**ÒNine-Point Circle ConstructionÓ**

In this webpage we will examine the famous Nine Point
Circle by constructing it in GSP!

We will first begin by constructing any triangle ABC:

Connected with this triangle are three special
secondary triangles we can construct based on it.

The first is the **Medial** triangle.
We construct this triangle by connecting the midpoints of the sides of
the original triangle ABC. We will
color the **Medial** triangle red:

The second special triangle is the **Orthic** triangle.
This triangle is made by connecting the feet of the altitudes (or
perpendicular lines through each vertex of triangle ABC to the opposite
side).

First we construct the altitudes:

Then we connect the feet of the altitudes. We will color the **Orthic **triangle light blue:

The third special triangle is made by finding the
midpoints between the orthocenter and each vertex of triangle ABC. To review, the orthocenter is point of
concurrency of each of the altitudes, which we have already found:

Now we find the midpoints between each vertex of
triangle ABC and the orthocenter, like so:

Connecting these points forms the third special
triangle, which we will color green and call it the **Orthocenter**-**Midpoint**
triangle:

Combining all three special triangles, the **Medial** triangle, the **Orthic** triangle, and the **Orthocenter**-**Midpoint**
triangle, we get the following:

Now if we construct the circumcircle to each of these
special triangles, we observe something extraordinary! We find that each of these three
special triangles has the same circumcircle. In other words, the vertices of each special triangle lies
on the same circle:

I find this to be remarkable!

And no matter the shape of your triangle, it still
holds. Whether an isosceles
triangle, like so:

for example.

However, what happens if the triangle is obtuse, or a
right triangle. Some interesting
things occur.

Try and explore some more on your own!

Click here for a file
to manipulate.