Nine-point conic
In geometry, the nine-point conic of a complete quadrangle is a conic that passes through the three diagonal points and the six midpoints of sides of the complete quadrangle.
The nine-point conic was described by Maxime Bôcher in 1892. The better-known nine-point circle is an instance of Bôcher's conic. The nine-point hyperbola is another instance.
Bôcher used the four points of the complete quadrangle as three vertices of a triangle with one independent point:
The conic is an ellipse if lies in the interior of 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 is the orthocenter, one obtains the nine-point circle, and when is on the circumcircle of, then the conic is an equilateral hyperbola.
In 1912 Maud Minthorn showed that the nine-point conic is the locus of the center of a conic through four given points.
The nine-point conic with respect to a line is the conic through the six harmonic conjugates of the intersection of the sides of the complete quadrangle with.