Nine-point conic

[[File:Nine point conic.svg|right|thumb|400px| {{legend|blue|Four constituent points of the quadrangle ($A, B, C, P$)}}

If $P$ were inside triangle $△ABC$, the nine-point conic would be a nine-point circle.]]

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:
 * Given a triangle $△ABC$ and a point $AC$ 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, and
 * the 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.

In 1912 Maud Minthorn showed that the nine-point conic is the locus of the center of a conic through four given points.