An Exploration of the Concurrency Points in a Triangle 

 

  Contents

  Problem Statement

  Approach

  Group Discussion

   Related Resource:

http://www.cs.princeton.edu/~ah/alg_anim/version2/NinePtCircle.html 
 


Problem Statement

Using analytic geometry and as much matrix notation as you can employ, show that the nine-point circle and the circumcircle are related by a linear transformation, a dilation using the orthocenter as the center of the dilation and a scaling factor of 1/2. The transformation matrix for a dilation with center (c1, c2, 1) and ratio r is 

 

Approach 

First of all, we used the orthocenter as our center for dialation. In our equation, this was the point: 

 

 

Since the scaling factor was ½, that is what we plugged in for r in the transformation matrix. Next, we multiplied the transformation matrix, 

 

 

by a set of three points 

 

 

which are the points that are the vertices of the triangle that intersect the circumcircle. This gave us three image points on the nine point circle. They are of the form 

 

 

This matrix would seem logical since we are looking at the midpoint of the orthocenter (p,q) and the points on the circumcircle (X,Y). Their sum divided by 2 illustrates the midpoint of the segment.    

 

We used the three points A B and C from the picture above. 

A=           B=          C=

 

 

Then we multiplied the transformation matrix by the set of these three points. We came up with the image points below. 

 

A’=        B’=       C’=

 

Group Discussion

We found it interesting that the circumcircle dilated around the orthocenter of the triangle that created the nine point circle. The dilation making the circle shrink down more in some places than others was also an interesting discovery. We understand this more thoroughly after our investigation since we looked into the idea of the midpoints as stated in our strategic plan. They are like the matrix above with the points divided by two, hence the definition of midpoint.