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. It is a fairly straight forward exercise to construct these nine points in Geometer's Sketchpad.
Nine Points Geometer's Sketchpad Script. (select New Sketch from the file menu, plot and select three points on the page and then run the script)
In order to construct the circle through the nine points in the above script one must find the center of that circle. This can be accomplished by constructing any of the interior triangles and then constructing the circumcircle for that triangle.
Using the Medial Triangle (the triangle formed by connecting the midpoints of each of the sides of our triangle): The Circumcenter (the point of concurrence of the perpendicular bisectors of the sides of the triangle) of the Medial Triangle is the center of the nine point circle.
Medial Triangle Construction of the Nine Point Circle.
Try using the Orthic Triangle (the triangle formed by connecting the points at the feet of the altitudes).
Try using the Orthocenter, Mid-Segment Triangle (the triangle formed by the points that are the midpoints of the segments from the vertices to the orthocenter).
In fact, we can use the Circumcenter of any triangle formed by choosing any three of the nine points in order to find the center of our nine point circle. This is true because the perpendicular bisector of a chord contains the center of the circle. Furthermore, two lines intersect in exactly one point. Thus, the perpendicular bisectors of any two chords of the circle must intersect at the circle's center.
We might also try looking at the relationship of the centroid and the orthocenter of our triangle. See what happens when we find the midpoint between the centroid and the orthocenter and try to explain this relationship.
What happens when we line up the three midpoints of the sides of our triangle and the three points at the feet of the altitudes? Make some conjecture and try to prove it.