Nine-point conic

{{short description|Geometric curve associated with a quadrangle}}

[[File:Nine point conic.svg|right|thumb|400px|

{{legend|blue|Four constituent points of the quadrangle ({{mvar|A, B, C, P}})}}

{{legend-line|solid #a17123|Six constituent lines of the quadrangle}}

{{legend-line|solid #188f61|Nine-point conic (a nine-point hyperbola, since {{mvar|P}} is across side {{mvar|AC}})}}

If {{mvar|P}} were inside triangle {{math|△ABC}}, the nine-point conic would be an ellipse.]]

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) [https://www.jstor.org/stable/1967142 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 {{math|△ABC}} and a point {{mvar|P}} in its plane, a conic can be drawn through the following nine points:

:: the midpoints of the sides of {{math|△ABC}},

:: the midpoints of the lines joining {{mvar|P}} to the vertices, and

:: the points where these last named lines cut the sides of the triangle.

The conic is an ellipse if {{mvar|P}} lies in the interior of {{math|△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 {{mvar|P}} is the orthocenter, one obtains the nine-point circle, and when {{mvar|P}} is on the circumcircle of {{math|△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.Maud A. Minthorn (1912) [http://babel.hathitrust.org/cgi/pt?id=uc1.b3808276;view=1up;seq=1 The Nine Point Conic], Master's dissertation at University of California, Berkeley, link from HathiTrust.

The nine-point conic with respect to a line {{math|l}} is the conic through the six harmonic conjugates of the intersection of the sides of the complete quadrangle with {{math|l}}.