The nine-point circle

This diagram shows the famous "nine point circle" and Euler's line. The centres are:
  • The Circumcentre (O)
  • The Orthocentre (H)
  • The Centroid (G)
  • The nine-point centre (N)
The nine-point centre is the centre of a circle which goes through 9 points on the triangle: the mid-points of each side, the points where the altitudes of the triangle meet the opposite sides and the points mid-way between the circumcentre and each of the vertices.
Try moving the points A, B and C. What do you notice about the points H, N, G and O? Can you prove that this is always the case?