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McCay-Griffiths Cubic

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Given a triangle and a point , the pedal triangle of is formed by the feet of the perpendiculars from to the sides . The circumcircle of the pedal triangle is called the pedal circle of (shown in black).

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The McCay–Griffiths cubic of is the set of all points such that the pedal circle of is tangent to the nine-point circle (shown in green) of . Its pivot point is the circumcenter (Kimberling center ).

Let , , be the side lengths; , , the parameters of in Conway triangle notation and , , the excenters of .

Then the equation of the McCay–Griffiths cubic of in barycentric coordinates is

,

where the sum is cyclic over all six permutations of , , .

The McCay–Griffiths cubic passes through the points , , and the Kimberling centers , , , , , , [1].

You can drag the vertices , , and the point .

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Contributed by: Minh Trinh Xuan (August 2022)
Open content licensed under CC BY-NC-SA

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References

[1] Encyclopedia of Triangle Centers (ETC). https://faculty.evansville.edu/ck6/encyclopedia/etc.html.

[2] B. Gilbert. "K003 McCay Cubic=Griffiths Cubic." (Jul 29, 2022) bernard-gibert.pagesperso-orange.fr/Exemples/k003.html.

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