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The Thomson cubic K002 and the Grebe cubic K102 are both isogonal pK with pivots G and K respectively. They generate a pencil of isogonal pK with pivot P on the line GK. This pencil contains only one circular cubic which is obtained when P = X(524). See also Table 32.

The table shows a selection of such cubics which all contain A, B, C, Ia, Ib, Ic, I, G, K (not mentioned in the table) and at least two more centers, very often P and its isogonal conjugate P*, a point on the circum-hyperbola through G and K with center X(1084).

P

cubic

centers on the cubic

X(2)

K002

X(3), X(4), X(9), X(57), X(223), X(282), X(1073), X(1249)

X(6)

K102

X(43), X(87), X(194)

X(69)

K169

X(20), X(25), X(64), X(69), X(159), X(200), X(269), X(1763), X(2138), X(2139)

X(81)

 

X(37), X(81)

X(86)

 

X(42), X(86)

X(141)

 

X(141), X(251), X(2896), X(2916)

X(183)

 

X(183), X(263)

X(298)

 

X(298), X(616)

X(299)

 

X(299), X(617)

X(302)

 

X(302), X(627)

X(303)

 

X(303), X(628)

X(323)

 

X(323), X(399), X(1138), X(1989)

X(325)

 

X(147), X(325), X(1976)

X(333)

 

X(10), X(19), X(58), X(63), X(333), X(573), X(1400)

X(385)

K128

X(32), X(76), X(98), X(385), X(511), X(694), X(1423), X(2319), X(3186)

X(394)

 

X(393), X(394), X(1422), X(1498), X(2324)

X(395)

K129a

X(14), X(16), X(18), X(62), X(395), X(1653), X(6151)

X(396)

K129b

X(13), X(15), X(17), X(61), X(396), X(1652), X(2981)

X(491)

 

X(487), X(491)

X(492)

 

X(488), X(492)

X(524)

 

X(111), X(524), X(2930) / the only circular cubic of the pencil

X(590)

 

X(588), X(590)

X(597)

K282

X(597)

X(599)

 

X(599), X(1383)

X(615)

 

X(589), X(615)

X(940)

 

X(940), X(941)

X(966)

 

X(966), X(967)

X(1150)

 

X(1150), X(2163)

X(1185)

 

X(1185), X(1218)

X(1211)

 

X(1169), X(1211)

X(1213)

 

X(1171), X(1213), X(1961)

X(1654)

 

X(1654), X(2248)

X(1812)

 

X(1812), X(1880)

X(1993)

 

X(155), X(254), X(1993), X(2165)

X(1994)

 

X(195), X(1994), X(2963)

X(2238)

 

X(1757), X(1929), X(2238), X(2664), X(2665), X(2238)*

X(2287)

 

X(28), X(72), X(610), X(1427), X(2184), X(2287)

X(2407)

 

X(2407), X(2433)

X(2421)

 

X(2395), X(2421)

X(3068)

K424a

X(371), X(485), X(493), X(3068)

X(3069)

K424b

X(372), X(486), X(494), X(3069)

P(5)

 

X(5), X(54), X(378)

P(7)

 

X(7), X(55), X(991)

P(8)

 

X(8), X(56), X(1764)

P(24)

 

X(24), X(30), X(68), X(74)

P(39)

K423

X(39), X(83), X(182), X(262)

 

 

 

 

 

 

Note : the points P(i) are not mentioned in ETC. Here are their first coordinates :

P(5) : 2*a^6 - 3*a^4*b^2 + b^6 - 3*a^4*c^2 - 2*a^2*b^2*c^2 - b^4*c^2 - b^2*c^4 + c^6

P(7) : a^4 - a^3*b - a^2*b^2 + a*b^3 - a^3*c - 3*a^2*b*c - a*b^2*c + b^3*c - a^2*c^2 - a*b*c^2 - 2*b^2*c^2 + a*c^3 + b*c^3

P(8) : -a^3 + a*b^2 - a*b*c + b^2*c + a*c^2 + b*c^2

P(24) : a^4*b^2 - 2*a^2*b^4 + b^6 + a^4*c^2 + 2*a^2*b^2*c^2 - b^4*c^2 - 2*a^2*c^4 - b^2*c^4 + c^6

P(39) : a^4 + 2*a^2*b^2 + 2*a^2*c^2 + b^2*c^2, harmonic conjugate of X(385) with respect to G and K