Every triangle has a special circle associated with it which passes through at most nine distinct points in the triangle: the midpoint of each side (labeled p1-p3), the foot of each of the three altitudes (labeled p4-p6), and the midpoint of each segment that connects a vertex to the orthocenter (labeled p7-p9).
As long as you are comfortable with the constructions of midpoints and the orthocenter, the construction is fairly straightforward. One trick is finding the center. We know the perpendicular bisectors of chords always go through the center of the circle. So I found the center by constructing two chords of the nine-point circle (by connecting two pairs of points with line segments), and then finding the point of intersection of the perpendicular bisectors of the chords, as shown in the figure below.
[ If you would like to use a script tool in GSP that constructs a nine-point circle, open this link and go to the tools menu. Note: you must have GSP on your computer to use the link. ]
As I moved the vertices of the triangle around and watched the nine-point circle change, I started to wonder how many different numbers of distinct points are possible. For example, do the nine-points ever collapse to just eight points? Here are some of the facts I found:
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