Triangle conic

The equation of a general triangle conic in trilinear coordinates {{math|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

$$\backslash begin\{align\}$$

& uyz + vzx + wxy = 0 \\[2pt]

& l^2 x^2 + m^2 y^2 + n^2 z^2 - 2mnyz - 2nlzx - 2lmxy = 0

\end{align}

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 {{math|△ABC}}. The angles at the vertices {{mvar|A, B, C}} are denoted by {{mvar|A, B, C}} and the lengths of the sides opposite to the vertices {{mvar|A, B, C}} are respectively {{mvar|a, b, c}}. The equations of the conics are given in the trilinear coordinates {{math|x : y : z}}. The conics are selected as illustrative of the several different ways in which a conic could be associated with a triangle.

File:Hofstadter.gif

An Hofstadter ellipse{{cite web |last1=Weisstein, Eric W. |title=Hofstadter Ellipse |url=https://mathworld.wolfram.com/HofstadterEllipse.html |website=athWorld--A Wolfram Web Resource. |publisher=Wolfram Research |access-date=25 November 2021}} is a member of a one-parameter family of ellipses in the plane of {{math|△ABC}} defined by the following equation in trilinear coordinates:

$$x^2\; +\; y^2\; +\; z^2\; +\; yz\backslash left[D(t)\; +\; \backslash frac\{1\}\{D(t)\}\backslash right]\; +\; zx\backslash left[E(t)\; +\; \backslash frac\{1\}\{E(t)\}\backslash right]\; +\; xy\backslash left[F(t)\; +\; \backslash frac\{1\}\{F(t)\}\backslash right]\; =\; 0$$

where {{mvar|t}} is a parameter and

$$\backslash begin\{align\}$$

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 {{mvar|t}} and {{math|1 − t}} are identical. When {{math|1=t = 1/2}} we have the inellipse

$$x^2+y^2+z^2\; -\; 2yz-\; 2zx\; -\; 2xy\; =0$$

and when {{math|t → 0}} we have the circumellipse

$$\backslash frac\{a\}\{Ax\}+\backslash frac\{b\}\{By\}+\backslash frac\{c\}\{Cz\}=0.$$

The family of Thomson conics consists of those conics inscribed in the reference triangle {{math|△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 {{math|△ABC}} such that the normals at the vertices of {{math|△ABC}} are concurrent. In both cases the points of concurrency lie on the Darboux cubic.{{cite journal |last1=Roscoe Woods |title=Some Conics with Names |journal=Proceedings of the Iowa Academy of Science |date=1932 |volume=39 Volume 50 |issue=Annual Issue}}{{cite web |title=K004 : Darboux cubic |url=https://bernard-gibert.pagesperso-orange.fr/Exemples/k004.html |website=Catalogue of Cubic Curves |publisher=Bernard Gibert |access-date=26 November 2021}}

File:EllipseOfParallelIntercepts.png

Given an arbitrary point in the plane of the reference triangle {{math|△ABC}}, if lines are drawn through {{mvar|P}} parallel to the sidelines {{mvar|BC, CA, AB}} intersecting the other sides at {{mvar|X_b, X_c, Y_c, Y_a, Z_a, Z_b}} 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 {{mvar|P}} are {{math|u : v : w}} the equation of the six-point conic is{{cite book |last1=Paul Yiu |title=Introduction to the Geometry of the Triangle |date=Summer 2001 |page=137 |url=http://math.fau.edu/Yiu/GeometryNotes020402.pdf |access-date=26 November 2021}}

$$-(au\; +\; bv\; +\; cw)^2(uyz\; +\; vzx\; +\; wxy)\; +\; (ax\; +\; by\; +\; cz)(vw(bv\; +\; cw)x\; +\; wu(cw\; +\; au)y\; +\; uv(au\; +\; bv)z)\; =\; 0$$

File:YffConics.gif

The members of the one-parameter family of conics defined by the equation

$$x^2+y^2+z^2-2\backslash lambda(yz+zx+xy)=0,$$

where $\backslash lambda$ is a parameter, are the Yff conics associated with the reference triangle {{math|△ABC}}.{{cite journal |last1=Clark Kimberling |title=Yff Conics |journal=Journal for Geometry and Graphics |date=2008 |volume=12 |issue=1 |pages=23–34}} A member of the family is associated with every point {{math|P(u : v : w)}} in the plane by setting

$$\backslash lambda=\backslash frac\{u^2+v^2+w^2\}\{2(vw+wu+uv)\}.$$

The Yff conic is a parabola if

$$\backslash lambda=\backslash frac\{a^2+b^2+c^2\}\{a^2+b^2+c^2-2(bc+ca+ab)\}=\backslash lambda\_0$$ (say).

It is an ellipse if and $\backslash lambda\_0\; >\; \backslash frac\{1\}\{2\}$ and it is a hyperbola if . For , the conics are imaginary.

• Triangle center
• Central line
• Triangle cubic
• Modern triangle geometry

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Category:Triangle geometry