center of the nine-point circle

The nine-point circle is a well-known circle associated with every triangle, passing through nine particular points: the midpoints of the three sides, the feet of the three altitudes, and the midpoints of the segments joining the orthocenter H to each of the three vertices. This circle is the image of the circumcircle under the homothety centered at G (the centroid) with ratio -1/2, and also under the homothety centered at H (the orthocenter) with ratio 1/2. Its center, denoted E, therefore lies on the Euler line, at the midpoint of the segment OH. Moreover, GE = (1/3) GH.