View source for McCay cubic

{{short description|Plane curve unique to a given triangle}}

In [[Euclidean geometry]], the '''McCay cubic''' (also called '''M'Cay cubic'''<ref>{{cite web |last1=Weisstein, Eric W |title=M'Cay Cubic |url=https://mathworld.wolfram.com/MCayCubic.html |website=MathWorld-A Wolfram Web Resource |publisher=Wolfram Research, Inc. |access-date=5 December 2021}}</ref> or '''Griffiths cubic'''<ref name="K003">{{cite web |last1=Bernard Gilbert |title=K003 McCay Cubic = Griffiths Cubic |url=https://bernard-gibert.pagesperso-orange.fr/Exemples/k003.html |website=Cubics in the Triangle Plane |publisher=Bernard Gilbert |access-date=5 December 2021}}</ref>) is a [[cubic plane curve]] in the plane of a reference [[triangle]] and associated with it. It is the third cubic curve in Bernard Gilbert's [[Catalogue of Triangle Cubics]] and it is assigned the identification number K003.<ref name="K003"/>

==Definition==
[[File:McCayCubicLocus.png|thumb|
{{legend-line|solid blue|Reference triangle {{math|△''ABC''}}}}
{{legend-line|solid magenta|[[Nine-point circle]] of {{math|△''ABC''}}}}
{{legend-line|solid brown|[[Pedal triangle]] of point {{mvar|P}}}}
{{legend-line|solid #03a9fc|Pedal circle ([[circumcircle]] of pedal triangle) of {{mvar|P}}}}
{{legend-line|solid red|'''McCay cubic''': [[Locus (mathematics)|locus]] of {{mvar|P}} such that the pedal circle and nine point circle are [[tangent]]}}
]]

The McCay cubic can be defined by [[Locus (mathematics)|locus]] properties in several ways.<ref name="K003"/> For example, the McCay cubic is the locus of a point {{mvar|P}} such that the pedal circle of {{mvar|P}} is [[tangent]] to the nine-point circle of the reference triangle {{math|△''ABC''}}.<ref>{{cite book |last1=John Griffiths |title=Mathematical Questions and Solutions from the Educational Times 2 (1902) 109, and 3 (1903) 29}}</ref> The McCay cubic can also be defined as the locus of point {{mvar|P}} such that the [[circumcevian triangle]] of {{mvar|P}} and {{math|△''ABC''}} are [[orthologic triangles|orthologic]].

==Equation of the McCay cubic==
The equation of the McCay cubic in [[barycentric coordinate system|barycentric coordinates]] <math>x:y:z</math> is
:<math>\sum_{\text{cyclic}}(a^2(b^2+c^2-a^2)x(c^2y^2-b^2z^2))=0.</math>
The equation in [[trilinear coordinates]] <math>\alpha : \beta : \gamma </math> is
:<math>\alpha (\beta^2 - \gamma^2)\cos A + \beta (\gamma^2 - \alpha^2)\cos B + \gamma (\alpha^2 - \beta^2)\cos C = 0</math>

==McCay cubic as a stelloid==
[[File:McCayStelloid.png|thumb|McCay cubic with its three concurring asymptotes]]
A stelloid is a cubic that has three real concurring [[asymptote]]s making 60° angles with one another. McCay cubic is a stelloid in which the three asymptotes concur at the [[centroid]] of triangle ABC.<ref name="K003"/> A circum-stelloid having the same asymptotic directions as those of McCay cubic and concurring at a certain (finite) is called '''McCay stelloid'''. The point where the asymptoptes concur is called the "radial center" of the stelloid.<ref>{{cite web |last1=Bernard Gilbert |title=McCay Stelloids |url=https://bernard-gibert.pagesperso-orange.fr/files/Resources/stelloidsK003.pdf |website=Catalogue of Triangle Cubics |publisher=Bernard Gilbert |access-date=25 December 2021}}</ref> Given a finite point X there is one and only one McCay stelloid with X as the radial center.

==References==
{{reflist}}

[[Category:Triangle geometry]]
[[Category:Curves defined for a triangle]]
[[Category:Eponymous curves]]