Triangle conic

The equation of a general triangle conic in trilinear coordinates 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} ==Perspector and dual conics==

The perspector of a circumconic or inconic is the perspector of the reference triangle and its polar triangle with respect to the conic. • A circumconic is the locus of trilinear poles of lines through its perspector. Conversely, the perspector of a circumconic lies on the trilinear polar of any point on the conic other than the triangle vertices. • The perspector of an inconic is its Brianchon point. A circumconic and an inconic are said to be dual if, using barycentric coordinates, coordinates of any point on the circumconic yield coefficients of an equation of a tangent to the inconic. • Pairs of dual conics include the Steiner ellipse and inellipse, and the Kiepert hyperbola and parabola. • Perspectors of dual conics are isotomic conjugates. • The dual circumconic of an inconic is the isotomic conjugate of the trilinear polar of its perspector. Note: Paris Pamfilos describes a different notion of dual conics by the property of sharing the same perspector. This notion also includes the Steiner ellipse and inellipse. Not all conics associated with a triangle are circumconics or inconics; for instance, the Artzt parabolas each only touch two vertices. ==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 . The angles at the vertices are denoted by and the lengths of the sides opposite to the vertices are respectively . The equations of the conics are given in the trilinear coordinates . The conics are selected as illustrative of the several different ways in which a conic could be associated with a triangle. Triangle circles Triangle ellipses Triangle hyperbolas . Triangle parabolas ==Families of triangle conics==

Hofstadter ellipses An Hofstadter ellipse is a member of a one-parameter family of ellipses in the plane of 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 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 and are identical. When we have the inellipse x^2+y^2+z^2 - 2yz- 2zx - 2xy =0 and when 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 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 such that the normals at the vertices of 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 , if lines are drawn through parallel to the sidelines intersecting the other sides at 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 first Lemoine circle. If the trilinear coordinates of are the equation of the six-point conic is -(au + bv + cw)^2(uyz + vzx + wxy) + (ax + by + cz)(vw(bv + cw)x + wu(cw + au)y + uv(au + bv)z) = 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 . A member of the family is associated with every point 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 and \lambda_0 > \frac{1}{2} and it is a hyperbola if \lambda_0 . For -1 , the conics are imaginary. ==See also==