Triangle conic explained

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 (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 has the form$$rx^2\; +\; sy^2\; +\; tz^2\; +\; 2uyz\; +\; 2vzx\; +\; 2wxy\; =\; 0.$$The equations of triangle circumconics and inconics have respectively the forms

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

Some well known triangle circles^[8]

No.	Name	Definition	Equation	Figure
1		Circle which passes through the vertices	(a)/(+ x) (b)/(+ y) (c)/(z) =0
2		Circle which touches the sidelines internally	\pm\sqrt{x}\cos (A)/(2) \pm \sqrt{y}\cos (B)/(2) \pm \sqrt{z}\cos (C)/(2) = 0
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{align} \pm\sqrt{ - x}\cos (A)/(2) \pm \sqrt{y}\cos (B)/(2) \pm \sqrt{z}\cos (C)/(2) & = 0 \\[ 2 pt] \pm\sqrt{x}\cos (A)/(2) \pm \sqrt{ - y}\cos (B)/(2) \pm \sqrt{z}\cos (C)/(2) & = 0 \\[ 2 pt] \pm\sqrt{x}\cos (A)/(2) \pm \sqrt{y}\cos (B)/(2) \pm \sqrt{ - z}\cos (C)/(2) & = 0 \end{align}
4	Nine-point circle (or Feuerbach's circle, Euler's circle, Terquem's circle)		\begin{align} & x ^{2\sin} 2 A + y ^{2\sin} 2 B + z ^{2\sin} 2 C - \\ & 2 (yz \sin A + zx \sin B + xy \sin C) = 0 \end{align}
5	Lemoine circle	Draw lines through the Lemoine point (symmedian point) and parallel to the sides of triangle . The points where the lines intersect the sides lie on a circle known as the Lemoine circle.

Triangle ellipses

Some well known triangle ellipses

No.	Name	Definition	Equation	Figure
1		Conic passing through the vertices of and having centre at the centroid of	(1)/(+ ax) (1)/(+ by) (1)/(cz) =0
2		Ellipse touching the sidelines at the midpoints of the sides	\begin{align} &a ^{2} x ^{2} + b ^{2} y ^{2} + c ^{2} z ^{2} - \\ & 2 bcyz - 2 cazx - 2 abxy = 0 \end{align}

Triangle hyperbolas

Some well known triangle hyperbolas

No.	Name	Definition	Equation	Figure
1		If the three triangles,,, constructed on the sides of as bases, are similar, isosceles and similarly situated, then the lines concur at a point . The locus of is the Kiepert hyperbola.	(\sin(B - C))/(x) + (\sin(C - A))/(y) + (\sin(A - B))/(z) = 0
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{align} & (a(\sin 2 B - \sin 2 C))/(x) + (b(\sin 2 C - \sin 2 A))/(y) \\[ 2 pt] & + (c(\sin 2 A - \sin 2 B))/(z) = 0 \end{align}

Triangle parabolas

Some well known triangle parabolas

No.	Name	Definition	Equation	Figure
1	Artzt parabolas^[9]	A parabola which is tangent at to the sides and two other similar parabolas.	\begin{align} (x ^{2})/(a ^{2}) - (4 yz)/(bc) & = 0 \\[ 2 pt] (y ^{2})/(- b ^{2}) (4 zx)/(ca) & = 0 \\[ 2 pt] (z ^{2})/(- c ^{2}) (4 xy)/(ab) & = 0 \end{align}
2	Kiepert parabola^[10]	Let three similar isosceles triangles,, be constructed on the sides of . Then the envelope of the axis of perspectivity the triangles and is Kiepert's parabola.	\begin{align} & f ^{2} x ^{2} + g ^{2} y ^{2} + h ^{2} z ^{2} - \\[ 2 pt] & 2 fgxy - 2 ghyz - 2 hfzx = 0 , \\[ 8 pt] & where f = b ^{2} - c ^{2,} \ & g = c ^{2} - a ^{2,} h = a ^{2} - b ^{2. \end{align}}

Families of triangle conics

Hofstadter ellipses

An Hofstadter ellipse^[11] is a member of a one-parameter family of ellipses in the plane of defined by the following equation in trilinear coordinates:$$x^2\; +\; y^2\; +\; z^2\; +\; yz\backslash left[D(t)\; +\; \backslash frac\{1\}\{D(t)\}\backslash right]\; +\; zx\backslash left[E(t)\; +\; \backslash frac\{1\}\{E(t)\}\backslash right]\; +\; xy\backslash left[F(t)\; +\; \backslash frac\{1\}\{F(t)\}\backslash right]\; =\; 0$$where is a parameter andThe 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$$\backslash frac+\backslash frac+\backslash frac=0.$$

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.^{[12] [13]}

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 Lemoine circle. If the trilinear coordinates of are the equation of the six-point conic is^[14] $$-(u\; +\; v\; +\; w)^2(bcuyz\; +\; cavzx\; +\; abwxy)\; +\; (ax\; +\; by\; +\; cz)(vw(v\; +\; w)ax\; +\; wu(w\; +\; u)by\; +\; uv(u\; +\; v)cz)\; =\; 0$$

Yff conics

The members of the one-parameter family of conics defined by the equation $$x^2+y^2+z^2-2\backslash lambda(yz+zx+xy)=0,$$where

λ is a parameter, are the Yff conics associated with the reference triangle .^[15] A member of the family is associated with every point in the plane by setting$$\backslash lambda=\backslash frac.$$The Yff conic is a parabola if$$\backslash lambda=\backslash frac=\backslash lambda\_0$$ (say).It is an ellipse if λ < λ _{0} and λ _{0} > (1)/(2) and it is a hyperbola if λ _{0} < λ < -1 . For -1 < λ < (1)/(2) , the conics are imaginary.

See also

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

Notes and References

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