NinePointsConic

Nine Points Conic

Four points in general position {A₁, A₂, A₃, A₄} define a complete quadrilateral (q). Consider all conics (c) passing through these four points. The centers O of all these conics lie on another conic (d), passing through the 3 intersection points {C₁, C₂, C₃} of the opposite sides of the complete quadrilateral q and the 6 middles {M₁₂, M₁₃, ...} of the sides {A₁A₂, A₁A₃, ... } of q. The center Q of this conic is the common middle point of the segments {M₁₃M₂₄, M₁₄M₂₃, M₃₄M₁₂}.

The conic (d) is a generalization of the Euler Circle of a triangle. The configuration here corresponds to the three vertices of the triangle plus its orthocenter as fourth point. In that case all the conics (c) are rectangular hyperbolas and their centers lie on the Euler circle of the triangle (see OrthoRectangular.html ).
In the general case, if q is non convex, then all conics (c) are hyperbolas and (d) is an ellipse. If q is convex then (d) is a hyperbola and the branch of (d) passing through the diagonal points of q is the locus of centers of hyperbolas, the other branch being the locus of centers of ellipses.
To obtain all points of (d) it suffices to consider all the conics passing through the four points {A₁, A₂, ...} and a fifth point B, varying on a small arc B₂B₃ of a circle. An alternative way to obtain all conics passing through the four points is to consider the family of conics generated by the two degenerate conics (c₁, c₂) represented by the pairs of lines intersecting at C₂ and C₃.
References
[AubertPapelier] Aubert, P & Papelier, G. Exercices de Geometrie Analytique (vol.I) Paris, Librairie Vuibert, 1966, p. 247.

Nine Points Conic

See Also