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 [1][2][3][4]). 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)".[5][6] 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.[7]

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{aligned}&uyz+vzx+wxy=0\\[2pt]&l^{2}x^{2}+m^{2}y^{2}+n^{2}z^{2}-2mnyz-2nlzx-2lmxy=0\end{aligned}}

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.

Triangle circles

Some well known triangle circles[8]
No.	Name	Definition	Equation	Figure
1	Circumcircle	Circle which passes through the vertices	${\frac {a}{x}}+{\frac {b}{y}}+{\frac {c}{z}}=0$	Circumcircle of $△ ABC$
2	Incircle	Circle which touches the sidelines internally	$\pm {\sqrt {x}}\cos {\frac {A}{2}}\pm {\sqrt {y}}\cos {\frac {B}{2}}\pm {\sqrt {z}}\cos {\frac {C}{2}}=0$	Incircle of $△ ABC$
3	Excircles (or escribed circles)	A circle lying outside the triangle, tangent to one of its sides and tangent to the extensions of the other two. Every triangle has three distinct excircles.	${\begin{aligned}\pm {\sqrt {-x}}\cos {\frac {A}{2}}\pm {\sqrt {y}}\cos {\frac {B}{2}}\pm {\sqrt {z}}\cos {\frac {C}{2}}&=0\\[2pt]\pm {\sqrt {x}}\cos {\frac {A}{2}}\pm {\sqrt {-y}}\cos {\frac {B}{2}}\pm {\sqrt {z}}\cos {\frac {C}{2}}&=0\\[2pt]\pm {\sqrt {x}}\cos {\frac {A}{2}}\pm {\sqrt {y}}\cos {\frac {B}{2}}\pm {\sqrt {-z}}\cos {\frac {C}{2}}&=0\end{aligned}}$	Incircle and excircles
4	Nine-point circle (or Feuerbach's circle, Euler's circle, Terquem's circle)	Circle passing through the midpoint of the sides, the foot of altitudes and the midpoints of the line segments from each vertex to the orthocenter	${\begin{aligned}&x^{2}\sin 2A+y^{2}\sin 2B+z^{2}\sin 2C\ -\\&2(yz\sin A+zx\sin B+xy\sin C)=0\end{aligned}}$	The nine points
5	Lemoine circle	Draw lines through the Lemoine point (symmedian point) $K$ and parallel to the sides of triangle $△ ABC$ . The points where the lines intersect the sides lie on a circle known as the Lemoine circle.		Lemoine circle of triangle ABC

Triangle ellipses

Some well known triangle ellipses
No.	Name	Definition	Equation	Figure
1	Steiner ellipse	Conic passing through the vertices of $△ ABC$ and having centre at the centroid of $△ ABC$	${\frac {1}{ax}}+{\frac {1}{by}}+{\frac {1}{cz}}=0$	Steiner ellipse of $△ ABC$
2	Steiner inellipse	Ellipse touching the sidelines at the midpoints of the sides	${\begin{aligned}&a^{2}x^{2}+b^{2}y^{2}+c^{2}z^{2}-\\&2bcyz-2cazx-2abxy=0\end{aligned}}$	Steiner inellipse of $△ ABC$

Triangle hyperbolas

Some well known triangle hyperbolas
No.	Name	Definition	Equation	Figure
1	Kiepert hyperbola	If the three triangles $△ XBC$ , $△ YCA$ , $△ ZAB$ , constructed on the sides of $△ ABC$ as bases, are similar, isosceles and similarly situated, then the lines $AX, BY, CZ$ concur at a point $N$ . The locus of $N$ is the Kiepert hyperbola.	${\frac {\sin(B-C)}{x}}+{\frac {\sin(C-A)}{y}}+{\frac {\sin(A-B)}{z}}=0$	Kiepert hyperbola of $△ ABC$ . The hyperbola passes through the vertices $A, B, C$ , the orthocenter ( $O$ ) and the centroid ( $G$ ) of the triangle.
2	Jerabek hyperbola	The conic which passes through the vertices, the orthocenter and the circumcenter of the triangle of reference is known as the Jerabek hyperbola. It is always a rectangular hyperbola.	${\begin{aligned}&{\frac {a(\sin 2B-\sin 2C)}{x}}+{\frac {b(\sin 2C-\sin 2A)}{y}}\\[2pt]&+{\frac {c(\sin 2A-\sin 2B)}{z}}=0\end{aligned}}$	Jerabek hyperbola of $△ ABC$

Triangle parabolas

Some well known triangle parabolas
No.	Name	Definition	Equation	Figure
1	Artzt parabolas[9]	A parabola which is tangent at $B, C$ to the sides $AB, AC$ and two other similar parabolas.	${\begin{aligned}{\frac {x^{2}}{a^{2}}}-{\frac {4yz}{bc}}&=0\\[2pt]{\frac {y^{2}}{b^{2}}}-{\frac {4zx}{ca}}&=0\\[2pt]{\frac {z^{2}}{c^{2}}}-{\frac {4xy}{ab}}&=0\end{aligned}}$	Artzt parabolas of $△ ABC$
2	Kiepert parabola[10]	Let three similar isosceles triangles $△ A'BC$ , $△ AB'C$ , $△ ABC'$ be constructed on the sides of $△ ABC$ . Then the envelope of the axis of perspectivity the triangles $△ ABC$ and $△ A'B'C'$ is Kiepert's parabola.	${\begin{aligned}&f^{2}x^{2}+g^{2}y^{2}+h^{2}z^{2}-\\[2pt]&2fgxy-2ghyz-2hfzx=0,\\[8pt]&{\text{where }}f=b^{2}-c^{2},\\&g=c^{2}-a^{2},\ h=a^{2}-b^{2}.\end{aligned}}$	Kiepert parabola of $△ ABC$ . The figure also shows a member (line $LMN$ ) of the family of lines whose envelope is the Kiepert parabola.

Families of triangle conics

Hofstadter ellipses

Family of Hofstadter conics of

△ ABC

An Hofstadter ellipse[11] 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 $t$ is a parameter and

{\begin{aligned}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{aligned}}

The ellipses corresponding to $t$ and $1 - t$ are identical. When $t = 1/2$ we have the inellipse

x^{2}+y^{2}+z^{2}-2yz-2zx-2xy=0

and when $t \to 0$ 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.[12][13]

Conic associated with parallel intercepts

Conics associated with parallel intercepts

Given an arbitrary point in the plane of the reference triangle $△ ABC$ , if lines are drawn through $P$ parallel to the sidelines $BC, CA, AB$ intersecting the other sides at $X b, X c, Y c, Y a, Z a, Z b$ 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 $P$ are $u : v : w$ the equation of the six-point conic is[14]

-(u+v+w)^{2}(bcuyz+cavzx+abwxy)+(ax+by+bz)(vw(v+w)bx+wu(w+u)by+uv(u+v)cz)=0

Yff conics

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 $△ ABC$ .[15] A member of the family is associated with every point $P (u : v : w)$ 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.

References

Paris Pamfilos (2021). "Equilaterals Inscribed in Conics". International Journal of Geometry. 10 (1): 5–24.
Christopher J Bradley. "Four Triangle Conics". Personal Home Pages. University of BATH. Retrieved 11 November 2021.
Gotthard Weise (2012). "Generalization and Extension of the Wallace Theorem". Forum Geometricorum. 12: 1–11. Retrieved 12 November 2021.
Zlatan Magajna. "OK Geometry Plus". OK Geometry Plus. Retrieved 12 November 2021.
"Geometrikon". Paris Pamfilos home page on Geometry, Philosophy and Programming. Paris Palmfilos. Retrieved 11 November 2021.
"1. Triangle conics". Paris Pamfilos home page on Geometry, Philosophy and Programming. Paris Palfilos. Retrieved 11 November 2021.
Bernard Gibert. "Catalogue of Triangle Cubics". Cubics in Triangle Plane. Bernard Gibert. Retrieved 12 November 2021.
Nelle May Cook (1929). A Triangle and its Circles (PDF). Kansas State Agricultural College. Retrieved 12 November 2021.
Nikolaos Dergiades (2010). "Conics Tangent at the Vertices to Two Sides of a Triangle". Forum Geometricorum. 10: 41–53.
R H Eddy and R Fritsch (June 1994). "The Conics of Ludwig Kiepert: A Comprehensive Lesson in the Geometry of the Tr". Mathematics Magazine. 67 (3): 188–205. doi:10.1080/0025570X.1994.11996212.
Weisstein, Eric W. "Hofstadter Ellipse". athWorld--A Wolfram Web Resource. Wolfram Research. Retrieved 25 November 2021.
Roscoe Woods (1932). "Some Conics with Names". Proceedings of the Iowa Academy of Science. 39 Volume 50 (Annual Issue).
"K004 : Darboux cubic". Catalogue of Cubic Curves. Bernard Gibert. Retrieved 26 November 2021.
Paul Yiu (Summer 2001). Introduction to the Geometry of the Triangle (PDF). p. 137. Retrieved 26 November 2021.
Clark Kimberling (2008). "Yff Conics". Journal for Geometry and Graphics. 12 (1): 23–34.

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[1] Paris Pamfilos (2021). "Equilaterals Inscribed in Conics". International Journal of Geometry. 10 (1): 5–24.

[2] Christopher J Bradley. "Four Triangle Conics". Personal Home Pages. University of BATH. Retrieved 11 November 2021.

[3] Gotthard Weise (2012). "Generalization and Extension of the Wallace Theorem". Forum Geometricorum. 12: 1–11. Retrieved 12 November 2021.

[4] Zlatan Magajna. "OK Geometry Plus". OK Geometry Plus. Retrieved 12 November 2021.

[5] "Geometrikon". Paris Pamfilos home page on Geometry, Philosophy and Programming. Paris Palmfilos. Retrieved 11 November 2021.

[6] "1. Triangle conics". Paris Pamfilos home page on Geometry, Philosophy and Programming. Paris Palfilos. Retrieved 11 November 2021.

[7] Bernard Gibert. "Catalogue of Triangle Cubics". Cubics in Triangle Plane. Bernard Gibert. Retrieved 12 November 2021.

[8] Nelle May Cook (1929). A Triangle and its Circles (PDF). Kansas State Agricultural College. Retrieved 12 November 2021.

[9] Nikolaos Dergiades (2010). "Conics Tangent at the Vertices to Two Sides of a Triangle". Forum Geometricorum. 10: 41–53.

[10] R H Eddy and R Fritsch (June 1994). "The Conics of Ludwig Kiepert: A Comprehensive Lesson in the Geometry of the Tr". Mathematics Magazine. 67 (3): 188–205. doi:10.1080/0025570X.1994.11996212.

[11] Weisstein, Eric W. "Hofstadter Ellipse". athWorld--A Wolfram Web Resource. Wolfram Research. Retrieved 25 November 2021.

[12] Roscoe Woods (1932). "Some Conics with Names". Proceedings of the Iowa Academy of Science. 39 Volume 50 (Annual Issue).

[13] "K004 : Darboux cubic". Catalogue of Cubic Curves. Bernard Gibert. Retrieved 26 November 2021.

[14] Paul Yiu (Summer 2001). Introduction to the Geometry of the Triangle (PDF). p. 137. Retrieved 26 November 2021.

[15] Clark Kimberling (2008). "Yff Conics". Journal for Geometry and Graphics. 12 (1): 23–34.