Orthocentroidal circle

In geometry, the orthocentroidal circle of a non-equilateral triangle is the circle that has the triangle's orthocenter and centroid at opposite ends of its diameter.

This diameter also contains the triangle's nine-point center and is a subset of the Euler line, which also contains the circumcenter outside the orthocentroidal circle.

Andrew Guinand showed in 1984 that the triangle's incenter must lie in the interior of the orthocentroidal circle, but not coinciding with the nine-point center; that is, it must fall in the open orthocentroidal disk punctured at the nine-point center.

451–452 The incenter could be any such point, depending on the specific triangle having that particular orthocentroidal disk.

The set of potential locations of one or the other of the Brocard points is also the open orthocentroidal disk.

A triangle (black), its orthocenter (blue), its centroid (red), and its orthocentroidal disk (yellow)
Orthocentroidal circle bounded by the orthocenter (H) and centroid (S)
Euler line , on which the circumcenter (O) and nine-point center (N) both lie along with H and S
F 1 and F 2 : Fermat points