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Given an isoconjugation with pole W = (p : q : r) swapping M and M*, the locus of M such that the trilinear polars of M and M* are parallel is the pivotal isocubic whose pivot is the pole W. It always contains the centroid G and the tangents at A, B, C are the medians. W and W/G (cevian quotient) are two other points of the cubic. See Special Isocubics ยง1.4.2.

The points at infinity of pK(W,W) are those of pK(G,W') where W' is the isotomic conjugate of W.

The isotomic transform of pK(W,W) is the cubic pK(W',G). See CL048.

The general equation of these cubics is :

The following table gives a selection of pK(W,W).

W

cubic or X(i) on the cubic

remark

X(1)

K101

 

X(2)

union of the medians

 

X(4)

K181

 

X(6)

K102

isogonal pK

X(67)

K103

the only circular pK(W,W)

X(69)

X(2), X(20), X(69)

isotomic transform of K663

X(75)

X(2), X(8), X(75), X(85)

isotomic transform of K363

X(76)

K184

isotomic transform of K002

E(387)

K104

the only equilateral pK(W,W)

X(34)

X(1), X(2), X(34), X(87), X(278), X(1722)

 

X(56)

X(1), X(2), X(56), X(57), X(87), X(509), X(978), X(1423)

 

X(264)

X(2), X(4), X(76), X(264), X(491), X(492)

isotomic transform of K168

X(274)

X(2), X(75), X(85), X(86), X(274), X(333), X(348)

isotomic transform of K345

X(276)

X(2), X(76), X(95), X(264), X(275), X(276)

isotomic transform of K612

X(286)

X(2), X(27), X(29), X(75), X(85), X(286)

 

X(290)

X(2), X(76), X(98), X(264), X(287), X(290), X(511)

isotomic transform of K357, see CL032

X(292)

K136

 

X(300)

K342a

isotomic transform of K341a

X(301)

K342b

isotomic transform of K341b

X(308)

X(2), X(76), X(83), X(264), X(308), X(1799)

isotomic transform of K836

X(313)

X(2), X(10), X(76), X(264), X(306), X(313), X(1330)

 

X(327)

X(2), X(76), X(262), X(264), X(327), X(1352)

 

X(331)

X(2), X(75), X(85), X(92), X(273), X(331)

 

X(334)

K868

isotomic transform of K251

X(349)

X(2), X(76), X(226), X(264), X(307), X(349)

 

X(393)

X(2), X(4), X(393), X(459), X(1123), X(1336)

 

X(870)

X(1), X(2), X(75), X(85), X(87), X(870)

 

X(1220)

X(1), X(2), X(10), X(87), X(894), X(1220)

 

X(1221)

X(2), X(37), X(75), X(85), X(1221), X(1909)

 

X(1434)

X(2), X(7), X(57), X(86), X(274), X(1434)

 

X(1494)

X(2), X(30), X(298), X(299), X(1494)

isotomic transform of K472

X(1502)

X(2), X(76), X(264), X(305), X(315), X(1502)

isotomic transform of K177

X(1989)

K278

 

X(2052)

X(2), X(92), X(1585), X(1586), X(2052)

isotomic transform of K857

X(2207)

X(2), X(6), X(19), X(393), X(1611), X(2207)

 

X(4590)

X(2), X(99), X(4590), X(6189), X(6190)

isotomic transform of K237

tX(44)

X(2), X(75), X(85), X(320), X(903)

isotomic transform of K453

tX(53)

X(2), X(69), X(95), X(276), X(394)

isotomic transform of K671

tX(187)

X(2), X(76), X(264), X(316), X(671)

isotomic transform of K043

tX(574)

X(2), X(76), X(264), X(598)

isotomic transform of K284

tX(1100)

X(2), X(75), X(85), X(319), X(321), X(1268)

isotomic transform of K637

tX(1691)

X(2), X(76), X(141), X(264), X(325), X(334), X(1916)

isotomic transform of K252

tX(2092)

X(2), X(76), X(86), X(264), X(274), X(314), X(1240)

isotomic transform of K253

tX(5019)

X(2), X(76), X(264)

isotomic transform of K321

tcE(371)

X(2)

isotomic transform of K485

Note : E(387) is the isotomic conjugate of Pt = E(371) = [2(b^4+c^4-2a^4)+2a^2(b^2+c^2)-b^2c^2 : ... : ...], intersection of the lines X(2)X(187) and X(30)X(76).

E(371) : SEARCH = 8.03922352257635; E(387) : SEARCH = 7.70234980923304, tcE(371) : SEARCH = 9.67361214847370.

tX(i) is the isotomic conjugate of X(i).

Remark :

Any pK(W,W) which passes through a point P also contains Q, the P-isoconjugate of the anticomplement of the isotomic conjugate of P. For example, if P = X(1) then Q = X(87). Generally, if P = u : v : w , then Q = u / (uv + uw - vw) : : .

Furthermore, pK(W,W) passes through P = u : v : w if and only if W lies on the circum-conic C(P) passing through P and its barycentric square P^2. This is the circum-conic with perspector S = u^2(v - w) : : , the intersection of the trilinear polar of P and the polar of P in the Steiner ellipse. C(P) is the isotomic transform of the line G-tP.