Triangle circles

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Some well known triangle circles^[8]

No.	Name	Definition	Equation	Figure
1	Circumcircle	Circle which passes through the vertices	${\displaystyle {\frac {a}{x}}+{\frac {b}{y}}+{\frac {c}{z}}=0}$
2	Incircle	Circle which touches the sidelines internally	${\displaystyle \pm {\sqrt {x}}\cos {\frac {A}{2}}\pm {\sqrt {y}}\cos {\frac {B}{2}}\pm {\sqrt {z}}\cos {\frac {C}{2}}=0}$
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.	${\displaystyle {\begin{aligned}\pm {\sqrt {-x}}\cos {\frac {A}{2}}\pm {\sqrt {y}}\cos {\frac {B}{2}}\pm {\sqrt {z}}\cos {\frac {C}{2}}&=0\\[2pt]\pm {\sqrt {x}}\cos {\frac {A}{2}}\pm {\sqrt {-y}}\cos {\frac {B}{2}}\pm {\sqrt {z}}\cos {\frac {C}{2}}&=0\\[2pt]\pm {\sqrt {x}}\cos {\frac {A}{2}}\pm {\sqrt {y}}\cos {\frac {B}{2}}\pm {\sqrt {-z}}\cos {\frac {C}{2}}&=0\end{aligned}}}$
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	${\displaystyle {\begin{aligned}&x^{2}\sin 2A+y^{2}\sin 2B+z^{2}\sin 2C\ -\\&2(yz\sin A+zx\sin B+xy\sin C)=0\end{aligned}}}$
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.

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Triangle ellipses

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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	${\displaystyle {\frac {1}{ax}}+{\frac {1}{by}}+{\frac {1}{cz}}=0}$
2	Steiner inellipse	Ellipse touching the sidelines at the midpoints of the sides	${\displaystyle {\begin{aligned}&a^{2}x^{2}+b^{2}y^{2}+c^{2}z^{2}-\\&2bcyz-2cazx-2abxy=0\end{aligned}}}$

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Triangle hyperbolas

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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.	${\displaystyle {\frac {\sin(B-C)}{x}}+{\frac {\sin(C-A)}{y}}+{\frac {\sin(A-B)}{z}}=0}$
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.	${\displaystyle {\begin{aligned}&{\frac {a(\sin 2B-\sin 2C)}{x}}+{\frac {b(\sin 2C-\sin 2A)}{y}}\\[2pt]&+{\frac {c(\sin 2A-\sin 2B)}{z}}=0\end{aligned}}}$

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Triangle parabolas

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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.	${\displaystyle {\begin{aligned}{\frac {x^{2}}{a^{2}}}-{\frac {4yz}{bc}}&=0\\[2pt]{\frac {y^{2}}{b^{2}}}-{\frac {4zx}{ca}}&=0\\[2pt]{\frac {z^{2}}{c^{2}}}-{\frac {4xy}{ab}}&=0\end{aligned}}}$
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.	${\displaystyle {\begin{aligned}&f^{2}x^{2}+g^{2}y^{2}+h^{2}z^{2}-\\[2pt]&2fgxy-2ghyz-2hfzx=0,\\[8pt]&{\text{where }}f=b^{2}-c^{2},\\&g=c^{2}-a^{2},\ h=a^{2}-b^{2}.\end{aligned}}}$

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Hofstadter ellipses

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:

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

where t is a parameter and

$${\displaystyle {\begin{aligned}D(t)&=\cos A-\sin A\cot tA\\E(t)&=\cos B-\sin B\cot tB\\F(t)&=\sin C-\cos C\cot tC\end{aligned}}}$$

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

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

and when t → 0 we have the circumellipse

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

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]

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]

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

Yff conics

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

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

where ${\displaystyle \lambda }$ 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

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

The Yff conic is a parabola if

$${\displaystyle \lambda ={\frac {a^{2}+b^{2}+c^{2}}{a^{2}+b^{2}+c^{2}-2(bc+ca+ab)}}=\lambda _{0}}$$

(say).

It is an ellipse if ${\displaystyle \lambda <\lambda _{0}}$ and ${\displaystyle \lambda _{0}>{\frac {1}{2}}}$ and it is a hyperbola if ${\displaystyle \lambda _{0}<\lambda <-1}$. For ${\displaystyle -1<\lambda <{\frac {1}{2}}}$, the conics are imaginary.