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.^[1] 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

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, 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.^[2]

The nine-point conic with respect to a line $l$ is the conic through the six harmonic conjugates of the intersection of the sides of the complete quadrangle with $l$ .