Nine-point conic

Short description: Geometric curve associated with a 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.

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 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