Triangle conic

{{short description|Conic plane curve associated with a given triangle}}

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 <ref>{{cite journal |last1=Paris Pamfilos |title=Equilaterals Inscribed in Conics |journal=International Journal of Geometry |date=2021 |volume=10 |issue=1 |pages=5–24}}</ref><ref>{{cite web |last1=Christopher J Bradley |title=Four Triangle Conics |url=https://people.bath.ac.uk/masgcs/ |website=Personal Home Pages |publisher=University of BATH|access-date=11 November 2021}}</ref><ref>{{cite journal |last1=Gotthard Weise |title=Generalization and Extension of the Wallace Theorem |journal=Forum Geometricorum |date=2012 |volume=12 |pages=1–11 |url=https://forumgeom.fau.edu/FG2012volume12/FG201201index.html |access-date=12 November 2021}}</ref><ref>{{cite web |last1=Zlatan Magajna |title=OK Geometry Plus |url=https://www.ok-geometry.com/binary/downloaddoc/id/14 |website=OK Geometry Plus |access-date=12 November 2021}}</ref>). However, Greek mathematician Paris Pamfilos defines a triangle conic as a "conic circumscribing a triangle {{math|△''ABC''}} (that is, passing through its vertices) or inscribed in a triangle (that is, tangent to its side-lines)".<ref>{{cite web |title=Geometrikon |url=http://users.math.uoc.gr/~pamfilos/eGallery/Gallery.html |website=Paris Pamfilos home page on Geometry, Philosophy and Programming |publisher=Paris Palmfilos |access-date=11 November 2021}}</ref><ref>{{cite web |title=1. Triangle conics |url=http://users.math.uoc.gr/~pamfilos/eGallery/problems/TriangleConics.html |website=Paris Pamfilos home page on Geometry, Philosophy and Programming |publisher=Paris Palfilos |access-date=11 November 2021}}</ref> 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.<ref>{{cite web |last1=Bernard Gibert |title=Catalogue of Triangle Cubics |url=https://bernard-gibert.pagesperso-orange.fr/ctc.html |website=Cubics in Triangle Plane |publisher=Bernard Gibert |access-date=12 November 2021}}</ref>

==Equations of triangle conics in trilinear coordinates== The equation of a general triangle conic in trilinear coordinates {{math|''x'' : ''y'' : ''z''}} has the form <math display=block>rx^2 + sy^2 + tz^2 + 2uyz + 2vzx + 2wxy = 0.</math> The equations of triangle circumconics and inconics have respectively the forms <math display=block>\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}</math>

==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.<ref>{{Cite web |title=Triangle conics |url=http://users.math.uoc.gr/~pamfilos/eGallery/problems/TriangleConics.html |website=Paris Pamfilos home page on Geometry, Philosophy and Programming |publisher=Paris Palfilos |access-date=11 October 2025}}</ref> Conversely, the perspector of a circumconic lies on the trilinear polar of any point on the conic other than the triangle vertices.<ref name=etc>{{Cite web | last = Kimberling | first = Clark |authorlink = Clark Kimberling | title = Encyclopedia of Triangle Centers | url = https://faculty.evansville.edu/ck6/encyclopedia/ETC.html#X11 | access-date = 11 October 2025}} See ''X''(11) = Feuerbach point, ''X''(101) = Ψ(incenter, symmedian point), ''X''(110) = Focus of Kiepert parabola, ''X''(115) = Center of Kiepert hyperbola, ''X''(125) = Center of Jerabek hyperbola, ''X''(190) = Yff parabolic point, ''X''(514) = Isogonal conjugate of ''X''(101), ''X''(523) = Isogonal conjugate of ''X''(110), ''X''(647) = Crossdifference of ''X''(2) and ''X''(3), and ''X''(650) = Crossdifference of ''X''(1) and ''X''(3).</ref> * The perspector of an inconic is its Brianchon point.<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Perspector |url=https://mathworld.wolfram.com/Perspector.html |website=MathWorld--A Wolfram Web Resource. |publisher=Wolfram Research |access-date=23 June 2025 |language=en}}</ref>

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.<ref name=yiu>{{cite book |last1=Paul Yiu |title=Introduction to the Geometry of the Triangle |date=Summer 2001 |pages=127, 133, 141 |url=https://users.math.uoc.gr/~pamfilos/Yiu.pdf |access-date=11 October 2025}}</ref> * The dual circumconic of an inconic is the isotomic conjugate of the trilinear polar of its perspector.<ref name=candi>{{Cite web |last=Stothers |first=Wilson |title=circumconics and inconics |url=https://www.maths.gla.ac.uk/wws/cabripages/misc/candi.htm |access-date=23 June 2025 |website=www.maths.gla.ac.uk}}</ref>

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.<ref>{{Cite web |title=Projectivities play related to a triangle |url=http://users.math.uoc.gr/~pamfilos/eGallery/problems/TriangleProjectivitiesPlay.html |website=Paris Pamfilos home page on Geometry, Philosophy and Programming |publisher=Paris Palfilos |access-date=23 June 2025}}</ref>

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 {{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.

===Triangle circles===

{| class="wikitable" |+ Some well known triangle circles<ref>{{cite book |last1=Nelle May Cook |title=A Triangle and its Circles |date=1929 |publisher=Kansas State Agricultural College |url=https://krex.k-state.edu/dspace/bitstream/handle/2097/23902/LD2668T41929C65.pdf?sequence=1&isAllowed=y |access-date=12 November 2021}}</ref> |- ! No. !! Name!! Definition !! Equation !! Figure |- | 1 || Circumcircle || Circle which passes through the vertices || style="text-align: center;" | <math> \frac{a}{x}+\frac{b}{y}+\frac{c}{z}=0</math> || thumb|Circumcircle of {{math|△''ABC''}} |- | 2 || Incircle || Circle which touches the sidelines internally || style="text-align: center;" | <math>\pm\sqrt{x}\cos\frac{A}{2} \pm \sqrt{y}\cos\frac{B}{2} \pm \sqrt{z}\cos\frac{C}{2} = 0</math> || thumb|Incircle of {{math|△''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. || style="text-align: center;" | <math>\begin{align} \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{align}</math> || right|thumb|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 || style="text-align: center;" | <math>\begin{align} & x^2\sin 2A + y^2\sin 2B + z^2\sin 2C \ - \\ & 2(yz \sin A + zx \sin B + xy \sin C) = 0 \end{align}</math> || thumb|The nine points |- | 5 || Polar circle || Circle centered at the orthocenter {{mvar|H}} with respect to which {{math|△''ABC''}} is self-polar: opposite vertices and sides are corresponding pole-polar pairs. The polar circle is real if and only if {{math|△''ABC''}} is obtuse. || || thumb|Polar circle of {{math|△''ABC''}}, centered at {{mvar|H}} |- | 6 || Orthocentroidal circle || Circle with the line segment joining the orthocenter {{mvar|H}} to the centroid {{mvar|G}} as a diameter. The orthocentroidal circle is the inverse of the orthic axis in the polar circle.<ref name=gibert-brocard>{{Cite web |last=Gibert |first=Bernard |title=Brocard triangles |url=http://bernard-gibert.fr/gloss/brocardtriangles.html |website=Cubics in the Triangle Plane |access-date=10 January 2026 |language=en}} "Recall that the orthoassociate (i.e. the inverse in the polar circle) of the orthic axis is the orthocentroidal circle."</ref>The intersections of the orthocentroidal circle and the altitudes form a triangle similar to {{math|△''ABC''}} which shares the same symmedian point {{mvar|K}}.<ref>{{Cite web | last = Kimberling | first = Clark |authorlink = Clark Kimberling | title = Encyclopedia of Triangle Centers Part 4 | url = https://faculty.evansville.edu/ck6/encyclopedia/ETCPart4.html#X5476 | access-date = 10 January 2026}} See ''X''(5476) = Midpoint of ''X''(6) and ''X''(381).</ref> || || thumb|Orthocentroidal circle of {{math|△''ABC''}} with shaded interior |- | 7 || Brocard circle || Circle passing through the Brocard points with the line segment joining the symmedian point {{mvar|K}} to the circumcenter {{mvar|O}} as a diameter. The Brocard circle is the inverse of the Lemoine axis in the circumcircle.<ref>{{Cite web |date=23 November 2021 |title=Brocard points and triangles |page=5 |url=https://users.math.uoc.gr/~pamfilos/eGallery/problems/Brocard.pdf |website=Paris Pamfilos home page on Geometry, Philosophy and Programming |publisher=Paris Palfilos |access-date=10 January 2026}}</ref>The intersections of the Brocard circle and the perpendicular bisectors of the sides form a triangle similar to {{math|△''ABC''}} which shares the same centroid {{mvar|G}}.<ref name=gibert-brocard/> || || thumb|Brocard circle of {{math|△''ABC''}} |- | 8 || Cosine circle (or second Lemoine circle) || Draw lines through {{mvar|K}} antiparallel to the sides of {{math|△''ABC''}}. The six points where the lines intersect the sides lie on a circle known as the cosine circle, as the chord on each side is proportional to the cosine of the opposite angle. Its center is {{mvar|K}}.The six points form an inscribed hexagon with sides alternately antiparallel and parallel to the sides of {{math|△''ABC''}}. Such a hexagon is always cyclic and lies on a '''Tucker circle''', of which the cosine circle and circumcircle are special cases. All Tucker circles have center on the line {{mvar|KO}} (the Brocard axis).<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Tucker Circles |url=https://mathworld.wolfram.com/TuckerCircles.html |website=MathWorld--A Wolfram Web Resource. |publisher=Wolfram Research |access-date=11 October 2025 |language=en}}</ref>The six concyclic points that define a Tucker circle form two triangles similar to {{math|△''ABC''}}. The Miquel points of these triangles with respect to {{math|△''ABC''}} are the Brocard points {{mvar|Ω}} and {{mvar|Ω′}}. Every Tucker circle is concentric with a circle passing through the Brocard points and has radius <math>\frac{R}{\overline{O\varOmega}}</math> times larger, where {{mvar|R}} is the radius of the circumcircle.<ref>{{citation|last=Johnson|first=Roger A.|title=Advanced Euclidean Geometry|publisher=Dover Publ.|year=2007|orig-year=1929|pages=271-2,276}}</ref> || || thumb|Cosine circle of {{math|△''ABC''}} |- | 9 || First Lemoine circle (or triplicate-ratio circle) || Draw lines through {{mvar|K}} parallel to the sides of {{math|△''ABC''}}. The six points where the lines intersect the sides lie on a circle known as the first Lemoine circle.It is a Tucker circle with center at the midpoint of {{mvar|K}} and {{mvar|O}} and radius <math>\frac{R}{\overline{O\varOmega}}</math> times the radius of the Brocard circle. || || thumb|First Lemoine circle of {{math|△''ABC''}} |- |}

===Triangle ellipses===

{| class="wikitable" |+ Some well known triangle ellipses |- ! No. !! Name!! Definition !! Equation !! Figure |- | 1 || Steiner ellipse || Conic passing through the vertices of {{math|△''ABC''}} and having centre at the centroid of {{math|△''ABC''}} || style="text-align: center;" | <math>\frac{1}{ax}+\frac{1}{by}+\frac{1}{cz}=0</math>|| thumb|Steiner ellipse of {{math|△''ABC''}} |- | 2 || Steiner inellipse || Ellipse touching the sidelines at the midpoints of the sides || style="text-align: center;" | <math>\begin{align} &a^2 x^2 + b^2 y^2 + c^2 z^2 - \\ &2bcyz - 2cazx - 2abxy = 0 \end{align}</math> || thumb|Steiner inellipse of {{math|△''ABC''}} |- | 3 || Mandart inellipse || Ellipse touching the sidelines at the contact points of the excircles. Its center is the mittenpunkt and its perspector is the Nagel point. || style="text-align: center;" | <math>\begin{align} & f^2 x^2 + g^2 y^2 + h^2 z^2 - \\[2pt] & 2fgxy - 2ghyz - 2 hfxz = 0, \\[8pt] & \text{where } f = \frac{a}{b+c-a}, \\ & g = \frac{b}{a+c-b}, \ h = \frac{c}{a+b-c}. \end{align}</math> || thumb|Mandart inellipse of {{math|△''ABC''}} |- |}

===Triangle hyperbolas===

{| class="wikitable" |+ Some well known triangle hyperbolas |- ! No. !! Name!! Definition !! Equation !! Figure |- | 1 || Kiepert hyperbola || If the three triangles {{math|△''XBC''}}, {{math|△''YCA''}}, {{math|△''ZAB''}}, constructed on the sides of {{math|△''ABC''}} as bases, are similar, isosceles and similarly situated, then the lines {{mvar|AX, BY, CZ}} concur at a point {{mvar|N}}. The locus of {{mvar|N}} is the Kiepert hyperbola.<ref name=kiepert>{{cite journal |last1=R H Eddy and R Fritsch |title=The Conics of Ludwig Kiepert: A Comprehensive Lesson in the Geometry of the Triangle |journal=Mathematics Magazine |date=June 1994 |volume=67 |issue=3 |pages=188–205|doi=10.1080/0025570X.1994.11996212}}</ref>The Kiepert hyperbola is rectangular and passes through the orthocenter and the centroid of {{math|△''ABC''}}. It is the isotomic conjugate of <math>GK</math>, the line joining the centroid and the symmedian point, and the isogonal conjugate of the Brocard axis. Its center is the inverse of the symmedian point in the orthocentroidal circle, the orthopole of the Brocard axis, the Steiner point of the medial triangle, and lies on the nine-point circle and the Steiner inellipse. Its perspector is the intersection of the orthic axis and the line at infinity.<ref name=etc/> || style="text-align: center;" | <math>\frac{\sin(B-C)}{x} + \frac{\sin(C-A)}{y} + \frac{\sin(A-B)}{z} = 0</math> || thumb|Kiepert hyperbola of {{math|△''ABC''}}. The hyperbola passes through the vertices {{mvar|A, B, C}}, the orthocenter ({{mvar|O}}) and the centroid ({{mvar|G}}) of the triangle. |- | 2 || Jerabek hyperbola || Rectangular hyperbola passing through the vertices, the orthocenter and the circumcenter of {{math|△''ABC''}}. Isogonal conjugate of the Euler line. Its center is the orthopole of the Euler line, the focus of the Kiepert parabola of the medial triangle, and lies on the nine-point circle and the orthic inconic. Its perspector is the intersection of the orthic axis and the Lemoine axis.<ref name=etc/> || style="text-align: center;" | <math>\begin{align} &\frac{a(\sin 2B - \sin 2C)}{x} + \frac{b(\sin 2C - \sin 2A)}{y} \\[2pt] &+ \frac{c(\sin 2A - \sin 2B)}{z} = 0 \end{align}</math>|| thumb|Jerabek hyperbola of {{math|△''ABC''}} |- | 3 || Feuerbach hyperbola || Rectangular hyperbola passing through the vertices, the orthocenter and the incenter of {{math|△''ABC''}}. Isogonal conjugate of <math>OI</math>, the line joining the circumcenter and the incenter. Its center is the orthopole of <math>OI</math>, the Feuerbach point, and lies on the incircle, the nine-point circle and the Mandart inellipse. Its perspector is the intersection of the orthic axis, the antiorthic axis and the Gergonne line.<ref name=etc/> || style="text-align: center;" | <math> \frac{\cos B - \cos C}x+ \frac{\cos C - \cos A}y+ \frac{\cos A - \cos B}z = 0</math> || thumb|Feuerbach Hyperbola of {{math|△''ABC''}} |- | 4 || Dual of the Yff parabola || Hyperbola passing through the vertices, the centroid and the Gergonne point of {{math|△''ABC''}}. Isotomic conjugate of the Nagel line.<ref name=yff>{{Cite web |last=Weisstein |first=Eric W. |title=Yff Parabola |url=https://mathworld.wolfram.com/YffParabola.html |website=MathWorld--A Wolfram Web Resource. |publisher=Wolfram Research |access-date=20 June 2025 |language=en}}</ref> Its center is the perspector of the Yff parabola of the medial triangle, and lies on the Steiner inellipse. Its perspector is the intersection of the Gergonne line and the line at infinity.<ref name=candi/><ref>{{Cite web | last = Kimberling | first = Clark |authorlink = Clark Kimberling | title = Encyclopedia of Triangle Centers Part 2 | url = https://faculty.evansville.edu/ck6/encyclopedia/ETCPart2.html#X1086 | access-date = 1 July 2025}} See ''X''(1086) = Center of hyperbola {{A,B,C,''X''(2),''X''(7)|}.</ref><ref name=etc/> || style="text-align: center;" | <math>\frac{bc(b-c)}{x}+\frac{ca(c-a)}{y}+\frac{ab(a-b)}{z} = 0</math> |- |}

'''Note:''' The pedal circle of any point on a rectangular circumhyperbola passes through the hyperbola's center. Since all such hyperbolas pass through the orthocenter, their centers all lie on the nine-point circle.<ref name=yiu></ref>

===Triangle parabolas===

{{anchor|Artzt parabolas}} {| class="wikitable" |+ Some well known triangle parabolas |- ! No. !! Name!! Definition !! Equation !! Figure |- | 1 || Artzt parabolas || A parabola (the {{mvar|A}}-Artzt parabola) tangent at {{mvar|B, C}} to the sides {{mvar|AB, AC}}, and two other similar parabolas.<ref>{{cite journal |last1=Nikolaos Dergiades |title=Conics Tangent at the Vertices to Two Sides of a Triangle |journal=Forum Geometricorum |date=2010 |volume=10 |pages=41–53}}</ref> The directrix of the {{mvar|A}}-Artzt parabola is the perpendicular to the median through {{mvar|A}} at its intersection, beside the midpoint of {{mvar|BC}}, with the nine-point circle. The focus of the {{mvar|A}}-Artzt parabola is the intersection of the symmedian through {{mvar|A}}, beside the symmedian point, with the Brocard circle.<ref>{{cite journal |last1=Sharp |first1=John |date=2 November 2015 |title=Artzt parabolas of a triangle |url=https://www.cambridge.org/core/journals/mathematical-gazette/article/abs/artzt-parabolas-of-a-triangle/685102E653A11FCA1FD50CD79F6552A2 |journal=The Mathematical Gazette |volume=99 |issue=546 |publisher=Cambridge University Press |pages=454, 458 |doi=10.1017/mag.2015.81 |access-date=11 October 2025|url-access=subscription }}</ref><ref>{{Cite web |date=15 May 2021 |title=Symmedian |url=http://users.math.uoc.gr/~pamfilos/eGallery/problems/Symmedian.pdf |website=Paris Pamfilos home page on Geometry, Philosophy and Programming |publisher=Paris Palfilos |access-date=24 June 2025}}</ref>|| style="text-align: center;" | <math> \begin{align} \frac{x^2}{a^2} - \frac{4yz}{bc} & = 0 \\[2pt] \frac{y^2}{b^2}-\frac{4xz}{ca} & = 0 \\[2pt] \frac{z^2}{c^2} -\frac{4xy}{ab} & = 0 \end{align}</math> || thumb|Artzt parabolas of {{math|△''ABC''}} |- | 2 || Kiepert parabola || Let three similar isosceles triangles {{math|△''A'BC''}}, {{math|△''AB'C''}}, {{math|△''ABC' ''}} be constructed on the sides of {{math|△''ABC''}}. Then the envelope of the perspectrix of the triangles {{math|△''ABC''}} and {{math|△''A'B'C' ''}} is Kiepert's parabola.<ref name=kiepert/>The Kiepert parabola has the Euler line as its directrix. Its focus (also called the '''Euler reflection point''') is the inverse of the centroid in the Brocard circle,<ref>{{Cite web | last = Cohl | first = Telv | title = Telv Cohl's Geometry Blog: Foci of Steiner inellipse and other triangle centers | url = https://artofproblemsolving.com/community/c284651h1706387_foci_of_steiner_inellipse_and_other_triangle_centers | website=AoPS | access-date = 9 January 2026}} See '''Property 2 :''' {{math|''E'', ''G'', ''T''}} are collinear and {{math|''E'' ↔ ''G''}} under the inversion WRT the Brocard circle {{math|⊙(''OK'')}} of {{math|△''ABC''}}.</ref> the trilinear pole of the Brocard axis, the orthocorrespondent of the center of the Kiepert hyperbola, the center of the Jerabek hyperbola of the anticomplementary triangle and, if {{math|△''ABC''}} is acute, the Feuerbach point of the tangential triangle. Its perspector is the Steiner point, which lies on the circumcircle and the Steiner ellipse.<ref name=etc/> || style="text-align: center;" | <math>\begin{align} & f^2 x^2 + g^2 y^2 + h^2 z^2 - \\[2pt] & 2fgxy - 2ghyz - 2 hfxz = 0, \\[8pt] & \text{where } f = b^2 - c^2, \\ & g = c^2 - a^2, \ h = a^2 - b^2. \end{align}</math> || thumb|Kiepert parabola of {{math|△''ABC''}}. The figure also shows a member (line {{mvar|LMN}}) of the family of lines whose envelope is the Kiepert parabola. |- | 3 || Yff parabola || Parabola tangent to the sides of {{math|△''ABC''}} whose directrix is the Brocard axis of the excentral triangle and whose focus is the center of the Kiepert hyperbola of the excentral triangle.<ref name=yff/> Its perspector is the trilinear pole of the Nagel line, and lies on the Steiner ellipse.<ref name=candi/><ref name=etc/> || style="text-align: center;" | <math>\begin{align} & f^2 x^2 + g^2 y^2 + h^2 z^2 - \\[2pt] & 2fgxy - 2ghyz - 2 hfxz = 0, \\[8pt] & \text{where } f = a^2(b-c)^2, \\ & g = b^2(c-a)^2, \ h = c^2(a-b)^2. \end{align}</math> || |- |}

==Families of triangle conics==

===Hofstadter ellipses=== thumb|Family of Hofstadter conics of {{math|△''ABC''}} An Hofstadter ellipse<ref>{{cite web |last1=Weisstein, Eric W. |title=Hofstadter Ellipse |url=https://mathworld.wolfram.com/HofstadterEllipse.html |website=MathWorld--A Wolfram Web Resource. |publisher=Wolfram Research |access-date=25 November 2021}}</ref> is a member of a one-parameter family of ellipses in the plane of {{math|△''ABC''}} defined by the following equation in trilinear coordinates: <math display=block>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</math> where {{mvar|t}} is a parameter and <math display=block>\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}</math> The ellipses corresponding to {{mvar|t}} and {{math|1 − ''t''}} are identical. When {{math|1=''t'' = 1/2}} we have the inellipse <math display=block>x^2+y^2+z^2 - 2yz- 2zx - 2xy =0</math> and when {{math|''t'' → 0}} we have the circumellipse <math display=block>\frac{a}{Ax}+\frac{b}{By}+\frac{c}{Cz}=0.</math>

===Conics of Thomson and Darboux=== 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.<ref>{{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}}</ref><ref>{{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}}</ref> thumb|Conic associated with parallel intercepts

===Conics associated with parallel intercepts=== 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|Xb, Xc, Yc, Ya, Za, Zb}} 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 {{mvar|P}} are {{math|''u'' : ''v'' : ''w''}} the equation of the six-point conic is<ref name=yiu></ref> <math display=block>-(au + bv + cw)^2(uyz + vzx + wxy) + (ax + by + cz)(vw(bv + cw)x + wu(cw + au)y + uv(au + bv)z) = 0</math>

===Yff conics=== thumb|Yff Conics The members of the one-parameter family of conics defined by the equation <math display=block>x^2+y^2+z^2-2\lambda(yz+zx+xy)=0,</math> where <math>\lambda</math> is a parameter, are the Yff conics associated with the reference triangle {{math|△''ABC''}}.<ref>{{cite journal |last1=Clark Kimberling |title=Yff Conics |journal=Journal for Geometry and Graphics |date=2008 |volume=12 |issue=1 |pages=23–34}}</ref> A member of the family is associated with every point {{math|''P''(''u'' : ''v'' : ''w'')}} in the plane by setting <math display=block>\lambda=\frac{u^2+v^2+w^2}{2(vw+wu+uv)}.</math> The Yff conic is a parabola if <math display=block>\lambda=\frac{a^2+b^2+c^2}{a^2+b^2+c^2-2(bc+ca+ab)}=\lambda_0</math> (say). It is an ellipse if <math>\lambda < \lambda_0</math> and <math>\lambda_0 > \frac{1}{2}</math> and it is a hyperbola if <math>\lambda_0 < \lambda < -1</math>. For <math> -1 < \lambda <\frac{1}{2}</math>, the conics are imaginary.

==References== {{reflist}}

Category:Triangle geometry