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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
The Nine Point Conic
Master's dissertation at University of California, Berkeley, link from
Some Considerations on the Nine-point Conic and its Reciprocal
Annals of Mathematics 8(6):185–8, link from Jstor. * Eric W. Weisstei
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 Bradle
The Nine-point Conic and a Pair of Parallel Lines
from University of Bath.
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.
a
Euclidean plane geometry Projective geometry
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
Maxime Bôcher (August 28, 1867 – September 12, 1918) was an American mathematician who published about 100 papers on differential equations, series, and algebra. He also wrote elementary texts such as ''Trigonometry'' and ''Analytic Geometry''. ...
in 1892. The better-known nine-point circle is an instance of Bôcher's conic. The nine-point hyperbola
In Euclidean geometry with triangle , the nine-point hyperbola is an instance of the nine-point conic described by American mathematician Maxime Bôcher in 1892. The celebrated nine-point circle is a separate instance of Bôcher's conic:
:Given ...
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
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. 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 (1912The Nine Point Conic
Master's dissertation at University of California, Berkeley, link from
HathiTrust
HathiTrust Digital Library is a large-scale collaborative repository of digital content from research libraries including content digitized via Google Books and the Internet Archive digitization initiatives, as well as content digitized locally ...
.
References
{{Reflist * Fanny Gates (1894Some Considerations on the Nine-point Conic and its Reciprocal
Annals of Mathematics 8(6):185–8, link from Jstor. * Eric W. Weisstei
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 Bradle
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
In academia and librarianship, conference proceedings is a collection of academic papers published in the context of an academic conference or workshop. Conference proceedings typically contain the contributions made by researchers at the conferen ...
25:38–41.
* Thomas F. Hogate (1894On 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
a
Euclidean plane geometry Projective geometry