geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, the nine-point conic of a

complete quadrangle In mathematics, specifically in incidence geometry and especially in projective geometry, a complete quadrangle is a system of geometric objects consisting of any four points in a plane, no three of which are on a common line, and of the six ...

is a

conic A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, thou ...

that passes through the three diagonal points and the six

midpoint In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment. Formula The midpoint of a segment in ''n''-dim ...

s of sides of the complete quadrangle. The nine-point conic was described by Maxime Bôcher in 1892. The better-known

nine-point circle In geometry, the nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant concyclic points defined from the triangle. These nine points are: * The midpoint of each s ...

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

s 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 In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...

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 In mathematics, a hyperbola is a type of smooth function, smooth plane curve, curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected component ( ...

. Bôcher notes that when is the

orthocenter The orthocenter of a triangle, usually denoted by , is the point (geometry), point where the three (possibly extended) altitude (triangle), altitudes intersect. The orthocenter lies inside the triangle if and only if the triangle is acute trian ...

, one obtains the nine-point circle, and when is on the

circumcircle In geometry, the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertex (geometry), vertices. The center of this circle is called the circumcenter of the triangle, and its radius is called the circumrad ...

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 The University of California, Berkeley (UC Berkeley, Berkeley, Cal, or California), is a Public university, public Land-grant university, land-grant research university in Berkeley, California, United States. Founded in 1868 and named after t ...

, link from

HathiTrust HathiTrust Digital Library is a large-scale collaborative repository of digital content from research libraries. Its holdings include content digitized via Google Books and the Internet Archive digitization initiatives, as well as content digit ...

. 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

{{Reflist * Fanny Gates (1894
Some Considerations on the Nine-point Conic and its Reciprocal

Annals of Mathematics The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as t ...

8(6):185–8, link from Jstor. * Eric W. Weisstei
Nine-point conic
from

MathWorld ''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science ...

. * 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 Taylor & Francis Group is an international company originating in the United Kingdom that publishes books and academic journals. Its parts include Taylor & Francis, CRC Press, Routledge, F1000 (publisher), F1000 Research and Dovepress. It i ...

publication. * Christopher Bradle
The Nine-point Conic and a Pair of Parallel Lines
from

University of Bath The University of Bath is a public research university in Bath, England. Bath received its royal charter in 1966 as Bath University of Technology, along with a number of other institutions following the Robbins Report. Like the University ...

External links

a

Euclidean plane geometry Projective geometry

References

Further reading

External links