The 9 Point Circle and its Center
The Nine-Point Circle for any triangle is the circle passing through the three midpoints of the sides of the triangle, the three feet of the altitudes of the triangle, and the three midpoints of the segments from the respective vertices to the orthocenter.
To construct this circle, we first construct D ABC. We then construct the medial triangle of D ABC. The vertices of this triangle are by definition the three midpoints of the sides of D ABC. We can label these points D, E, and F. Next, we construct the orthic triangle of D ABC. The vertices of this triangle are the feet of the altitudes of D ABC. We can label these points G, H, and I. Finally, we construct the orthocenter of D ABC, and then the mid-segment triangle with respect to this orthocenter. We can label these vertices J, K, and L. Note that the orthocenter of a triangle is normally labeled H, but it is convenient for this write-up to label it M.
So we have the following construction:
The nine vertices of these three inscribed triangles lie on the nine-point circle. We only need to locate the center of this circle to enable us to construct it.
If we construct the circumcircle of each of the inscribed triangles, we note that all three circumcircles overlay each other. We label the common circumcenter N. Since all nine points that we are interested in lie on any one of the circumcircles, we conclude that the center of the nine-point circle is in the same location as the circumcenter for any of the inscribed triangles.
So we now have the following construction:
We can now hide the three inscribed triangles and their circumcircles. We can then construct the nine-point circle using point N as the center and any of the vertices of the now hidden triangles as a point on the circle.
We finally have our nine-point circle:
Since we have already constructed the orthocenter (point M), if we continue and construct the centroid and circumcenter of D ABC, we can explore the relationship between these points and the center of the nine point circle for different shaped triangles. We will label the centroid O and the circumcenter P.
We note that a single line can be drawn through all 4 points (M, N, O, and P). Using GSP, we can see that for different acute triangles the nine point circle center is always inbetween the orthocenter and the centroid and is always on the opposite side of the centroid from the circumcenter.