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.Maxime Bôcher (1892) Nine-point Conic, Annals of Mathematics, link from Jstor. 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 and a point in its plane, a conic can be drawn through the following nine points: the midpoints of the sides of , the midpoints of the lines joining to the vertices, and the points where these last named lines cut the sides of the triangle. 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.Maud A. Minthorn (1912) The Nine Point Conic, Master's dissertation at University of California, Berkeley, link from HathiTrust.
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 .
References
Fanny Gates (1894) Some Considerations on the Nine-point Conic and its Reciprocal, Annals of Mathematics 8(6):185–8, link from Jstor. Eric W. Weisstein Nine-point conic from MathWorld. Michael DeVilliers (2006) The nine-point conic: a rediscovery and proof by computer from International Journal of Mathematical Education in Science and Technology, a Taylor & Francis publication. Christopher Bradley The Nine-point Conic and a Pair of Parallel Lines from University of Bath.
Further reading W. G. Fraser (1906) "On relations of certain conics to a triangle", Proceedings of the Edinburgh Mathematical Society 25:38–41. Thomas F. Hogate (1894) On the Cone of Second Order which is Analogous to the Nine-point Conic, Annals of Mathematics 7:73–6. P. Pinkerton (1905) "On a nine-point conic, etc.", Proceedings of the Edinburgh Mathematical Society 24:31–3.
External links Nine-point conic and Euler line generalization at Dynamic Geometry Sketches