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.

 I constructed the original triangle just using the segment tool.  You can drag any of the points to create different types of triangles.
 
 

Using the basic tools I then constructed the midpoints of all the sides and the perpendicular lines through each of the vertices and their opposite sides (Altitudes). The orthocenter is labeled, O, and the midpoints are labeled , D, E, and F.
 
 

Again using basic 'Construct' tools I constructed the midpoints of the segments created by each of the vertices and the Orthocenter, and I labeled the feet of the altitudes.  I now have the nine points.  From here, originally I just used the "Construct an arc given three points" function, but that didn't really give me the center of the Nine Point Circle.  So I did some minor research and found that the center is actually the midpoint of the segment created by the Orthocenter and the Circumcenter of a triangle. The Circumcenter is labeled, C.
 
 

The last thing to do is construct the circle using the center, O, and radius.  The thing that I find interesting at this point is that every time I constructed this diagram before, the circle would disappear on me whenever the triangle became obtuse.  However, you will notice that when you do drag one of the vertices, creating an obtuse angle, some of the points that make up the Nine Point Circle.  The error is in the construction of the original triangle.  Because segments were used, the feet of the altitudes cannot pass outside of the original triangle.  This is imperative, for in obtuse triangle the altitudes are actually formed outside of the triangle.

You will notice that in the picture below the problem has been solved.  I had to start all over again and create the original triangle using three intersecting lines. All Nine Points now remain no matter the type of triangle- Acute, Right, or Obtuse.