To start off this assignment, first construct any triangle ABC. Then construct the orthocenter of this triangle. Label it H.
Then construct three more triangle from the orthocenter to the vertices of the larger triangle.
Now construct the othocenter of the triangles just created.
Notice that this picture is the same as the first because orthocenter of these three triangle are simply the vertices of the triangle ABC. For the triangle HAB the orthocenter is the point C. For triangle HBC the orthocenter is point A. Then for the triangle HAC the orthocenter is the point B.
Now construct the circmcircle for all four triangles: ABC, HAB, HBC, and HAC.
Now lets construct the nine-point circle. To do this, construct the midpoint of the three segments from the vertices of the original triangle to its orthocenter. Then construct a triangle from these midpoints. Next construct the circumcircle of this triangle. This is the nine-point circle. The point G is the senter of this circle.
Now let's hide a few things and mark the points where the
nine-point circle crosses the elements of the triangle nine times.
The nine points of the nine-point circle are represented by the yellow points