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.

Since P is across side AC, the 9-point conic is a nine-point hyperbola in this instance. When P is inside triangle ABC the 9-point conic is 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.

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Further reading
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