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The pK with pole W = (p : q : r) and pivot P = (u : v : w) is a pK60+ if and only if P lies on the Neuberg cubic= K001. Its pole W lies on the cubic Co = K095. The locus of the common point of the asymptotes is the bicircular quartic Q004.

The correspondences between W and P are given by the formulas :

The table shows some remarkable examples of pK60+ with the intersection Q of the asymptotes on Q004.

pole W

pivot P

point Q

cubic

hessian or centers on the hessian

X(6)

X(3)

X(2)

McCay K003

McCay hessian cubic K048

X(53)

X(4)

X(51)

McCay orthic K049

 

X(395)

X(617)

X(619)

Fermat K046b

X(533), X(619)

X(396)

X(616)

X(618)

Fermat K046a

X(532), X(618)

X(1989)

X(30)

X(476)

Tixier K037

X(13), X(14), X(476), X(542)

X(2160)

X(1)

(2SA + bc + ca + ab)/(2SA+ bc) : :

K097

 

X(30)xX(1138)

X(1138)

X(3258)

Fermat K515

 

X(13)^2

X(13)

X(13)

see below

 

X(14)^2

X(14)

X(14)

see below

 

Note : when P is one of the Fermat points X(13) or X(14), the pK60+ decomposes into the cevian lines of P.

The hessian cubic of a pK60+ is always a focal cubic with singular focus the common point Q of the three asymptotes.