This "extra" exploration comes from Explorations 04: Centers of a Triangle

10. The Nine-Point circle for any triangle passes through the three mid-points of the sides, the three feet of the altitudes, and the three mid-points of the segments from the respective vertices to orthocenter. Construct the nine points, locate the center (N) and construct the nine point circle.

This construction will give you the step by step process on how to construct the nine point center and circle of any triangle; it also includes some interesting facts about the nine point circle.

Constructing the nine point circle and center:

1. Start with any triangle ABC

2. Find the midpoints of each side of the triangle.

3. Then, find the altitudes of the triangle by extending a perpendicular line from each of the vertices to the opposite sides of the triangle; the intersection point of the altitudes is also called the orthocenter. Create points where the perpendicular line intersects with the side of the triangle.

4. Create line segments from the orthocenter to each vertex of the triangle and find the midpoint of these segments; label the midpoints.

We now have the nine points that make up the circle, but we need a center in GSP to construct it.

5. Create the perpendicular bisector for each side of the triangle; the point where the perpendicular bisectors intersect is called the circumcenter.

6. Create a line segment connecting the orthocenter and the circumcenter, this is also called the Euler Line, and find the midpoint. The midpoint is the nine point circle center and now can be constructed using GSP.

The GSP Tool for creating the nine point circle and center is found in the GSP Script Tools in Assignment 5

The nine point circle for an obtuse triangle and an acute triangle:

Obtuse triangle: Notice the orthocenter and circumcenter lie on the outside of the triangle and some of the nine points also lie outside of the triangle.

Acute triangle: All significant points of the nine points circle lie on the triangle.

For a right triangle:

Some of the significant points of the nine point circle overlap the orthocenter and circumcenter.

Some interesting facts:

The radius of the nine point circle is half the radius of the circumcircle:

The nine point circle will bisect any segment from the orthocenter to any point on the circumcircle.

The nine point circle is tangent to the incircle:

And tangent to the excircles

Hope that was helpful in learning more about the nine point circle!