Nine-point hyperbola
In plane geometry with triangle ABC, the nine-point hyperbola is an instance of the nine-point conic described by Maxime Bôcher in 1892. The celebrated nine-point circle is a separate instance of Bôcher's conic: Given a triangle ABC and a point P in its plane, a conic can be drawn through the following nine points:the midpoints of the sides of ABC,the midpoints of the lines joining P to the vertices, andthe points where these last named lines cut the sides of the triangle. The conic is an ellipse if P lies in the interior of ABC or in one of the regions of the plane separated from the interior by two sides of the triangle; otherwise, the conic is a hyperbola. Bôcher notes that when P is the orthocenter, one obtains the nine-point circle, and when P is on the circumcircle of ABC, then the conic is an equilateral hyperbola. (Wikipedia).
