Thomas Earl Ricks

Mathematics Education

 

Assignment # 4

 

“Nine-Point Circle Construction”

 


 

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.

 


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