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.

  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 a nine-point circle.

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 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.

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