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.

[1] The better-known nine-point circle is an instance of Bôcher's conic.

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 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.

[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.

Four constituent points of the quadrangle ( A, B, C, P )
Six constituent lines of the quadrangle
Nine-point conic (a nine-point hyperbola , since P is across side AC )
If P were inside triangle △ ABC , the nine-point conic would be an ellipse.