Triangle conic

In Euclidean geometry, a triangle conic is a conic in the plane of the reference triangle and associated with it in some way. For example, the circumcircle and the incircle of the reference triangle are triangle conics. Other examples are the Steiner ellipse, which is an ellipse passing through the vertices and having its centre at the centroid of the reference triangle; the Kiepert hyperbola which is a conic passing through the vertices, the centroid and the orthocentre of the reference triangle; and the Artzt parabolas, which are parabolas touching two sidelines of the reference triangle at vertices of the triangle.

The terminology of triangle conic is widely used in the literature without a formal definition; that is, without precisely formulating the relations a conic should have with the reference triangle so as to qualify it to be called a triangle conic (see    ). However, Greek mathematician Paris Pamfilos defines a triangle conic as a "conic circumscribing a triangle $△ABC$ (that is, passing through its vertices) or inscribed in a triangle (that is, tangent to its side-lines)". The terminology triangle circle (respectively, ellipse, hyperbola, parabola) is used to denote a circle (respectively, ellipse, hyperbola, parabola) associated with the reference triangle is some way.

Even though several triangle conics have been studied individually, there is no comprehensive encyclopedia or catalogue of triangle conics similar to Clark Kimberling's Encyclopedia of Triangle Centres or Bernard Gibert's Catalogue of Triangle Cubics.

Equations of triangle conics in trilinear coordinates
The equation of a general triangle conic in trilinear coordinates $x : y : z$ has the form $$rx^2 + sy^2 + tz^2 + 2uyz + 2vzx + 2wxy = 0.$$ The equations of triangle circumconics and inconics have respectively the forms $$\begin{align} & uyz + vzx + wxy = 0 \\[2pt] & l^2 x^2 + m^2 y^2 + n^2 z^2 - 2mnyz - 2nlzx - 2lmxy = 0 \end{align}$$

Special triangle conics
In the following, a few typical special triangle conics are discussed. In the descriptions, the standard notations are used: the reference triangle is always denoted by $△ABC$. The angles at the vertices $A, B, C$ are denoted by $A, B, C$ and the lengths of the sides opposite to the vertices $A, B, C$ are respectively $a, b, c$. The equations of the conics are given in the trilinear coordinates $x : y : z$. The conics are selected as illustrative of the several different ways in which a conic could be associated with a triangle.

Hofstadter ellipses
An Hofstadter ellipse is a member of a one-parameter family of ellipses in the plane of $△ABC$ defined by the following equation in trilinear coordinates: $$x^2 + y^2 + z^2 + yz\left[D(t) + \frac{1}{D(t)}\right] + zx\left[E(t) + \frac{1}{E(t)}\right] + xy\left[F(t) + \frac{1}{F(t)}\right] = 0$$ where $K$ is a parameter and $$\begin{align} D(t) &= \cos A - \sin A \cot tA \\ E(t) &= \cos B - \sin B \cot tB \\ F(t) &= \sin C - \cos C \cot tC \end{align}$$ The ellipses corresponding to $AX, BY, CZ$ and $△ABC$ are identical. When $△ABC$ we have the inellipse $$x^2+y^2+z^2 - 2yz- 2zx - 2xy =0$$ and when $△ABC$ we have the circumellipse $$\frac{a}{Ax}+\frac{b}{By}+\frac{c}{Cz}=0.$$

Conics of Thomson and Darboux
The family of Thomson conics consists of those conics inscribed in the reference triangle $△ABC$ having the property that the normals at the points of contact with the sidelines are concurrent. The family of Darboux conics contains as members those circumscribed conics of the reference $△ABC$ such that the normals at the vertices of $△ABC$ are concurrent. In both cases the points of concurrency lie on the Darboux cubic.

Conics associated with parallel intercepts
Given an arbitrary point in the plane of the reference triangle $△XBC$, if lines are drawn through $N$ parallel to the sidelines $N$ intersecting the other sides at $A, B, C$ then these six points of intersection lie on a conic. If P is chosen as the symmedian point, the resulting conic is a circle called the Lemoine circle. If the trilinear coordinates of $O$ are $△YCA$ the equation of the six-point conic is $$-(u + v + w)^2(bcuyz + cavzx + abwxy) + (ax + by + cz)(vw(v + w)ax + wu(w + u)by + uv(u + v)cz) = 0$$

Yff conics
The members of the one-parameter family of conics defined by the equation $$x^2+y^2+z^2-2\lambda(yz+zx+xy)=0,$$ where $$\lambda$$ is a parameter, are the Yff conics associated with the reference triangle $△ZAB$. A member of the family is associated with every point $△ABC$ in the plane by setting $$\lambda=\frac{u^2+v^2+w^2}{2(vw+wu+uv)}.$$ The Yff conic is a parabola if $$\lambda=\frac{a^2+b^2+c^2}{a^2+b^2+c^2-2(bc+ca+ab)}=\lambda_0$$ (say). It is an ellipse if $$\lambda < \lambda_0$$ and $$\lambda_0 > \frac{1}{2}$$ and it is a hyperbola if $$\lambda_0 < \lambda < -1$$. For $$ -1 < \lambda <\frac{1}{2}$$, the conics are imaginary.