Nine-point center

It is so called because it is the center of the nine-point circle, a circle that passes through nine significant points of the triangle: the midpoints of the three edges, the feet of the three altitudes, and the points halfway between the orthocenter and each of the three vertices.

The centroid G also lies on the same line, 2/3 of the way from the orthocenter to the circumcenter,[2] so Thus, if any two of these four triangle centers are known, the positions of the other two may be determined from them.

[3][4][5][6] The distance from the nine-point center to the incenter I satisfies where R, r are the circumradius and inradius respectively.

[citation needed] The nine-point center lies at the centroid of four points: the triangle's three vertices and its orthocenter.

[8]: p.111 Of the nine points defining the nine-point circle, the three midpoints of line segments between the vertices and the orthocenter are reflections of the triangle's midpoints about its nine-point center.