Assignment # 4

by Shannon Umberger

### Introduction

In this write-up, I investigate the construction of the nine-point circle and prove that it exists. First, I look individually at the definitions and circumcircles of the medial triangle, orthic triangle, and orthocenter mid-segment triangle. Then, I put the three triangles together and hypothesize that their circumcircles coincide when the three triangles are constructed in the same given triangle. Finally, I call this common circumcircle the nine-point circle of the original triangle, and I prove that the hypothesis is true.

### Medial Triangle

The medial triangle of a given triangle is constructed by connecting the midpoints of the three sides of the given triangle. Take a look at the following example:

In this construction, point G is the centroid (intersection of the medians) of triangle ABC. The endpoints of these medians are the midpoints of sides AB, BC, and CA. Once these three midpoints are connected, the medial triangle is formed (in red).

The circumcircle of the medial triangle can then be constructed (in red). Point O1 is the center of this circle.

### Orthic Triangle

The orthic triangle of a given triangle is constructed by connecting the feet of the three altitudes of the given triangle. Take a look at the following example:

In this construction, point H is the orthocenter (intersection of the altitudes) of triangle ABC. The endpoints of these altitudes are constructed on sides AB, BC, and CA. Once these three points are connected, the orthic triangle is formed (in blue).

The circumcircle of the orthic triangle can then be constructed (in blue). Point O2 is the center of this circle.

### Orthocenter Mid-Segment Triangle

The orthocenter mid-segment triangle is constructed by connecting the midpoints of the three segments constructed by joining the orthocenter to each vertex of the original triangle. Take a look at the following example:

In this construction, point H is the orthocenter (intersection of the altitudes) of triangle ABC. The midpoints of segments HA, HB, and HC can be found. Once these midpoints are connected, the orthocenter mid-segment triangle is formed (in green).

The circumcircle of the orthocenter mid-segment triangle can then be constructed (in green). Point O3 is the center of this circle.

### Now, Put Them All Together...

If the reader has noticed, in each of the above figures, triangle ABC was always the exact same triangle. Now, take a look at the medial triangle, orthic triangle, and orthocenter mid-segment triangle all constructed together in triangle ABC...:

...and examine the circumcircles of those three triangles constructed together in triangle ABC:

Hopefully, the reader notices that the three circumcircles seem to coincide (above, in pink). Also, centers O1, O2, and O3 from the three original circumcircles seem to come together to form a single point, called point O in the above figure.

### The Nine-Point Circle

So, this conjecture remains: given a triangle, the midpoints of the sides of the triangle, the feet of the altitudes of the triangle, and the midpoints of the segments from the vertices of the triangle to its orthocenter, there exists a unique circle that contains these nine points.

Consequently, this conjecture is true, and the circle is called the nine-point circle. Can you prove it?