CL023

Let W = (p : q : r) be a pole of isoconjugation and P* the W-isoconjugate of point P.

The cevian triangle of P* and ABC are orthologic if and only if P lies on pK(W^2,W÷H) where W^2 is the barycentric square of W and W÷H = U is the barycentric quotient of W and H i.e. W^2 = (p^2 : q^2 : r^2) and U = (p SA : q SB : r SC).

Any such cubic is the transform of the Lucas cubic K007 in the isoconjugation with pole W.

pK(W^2,W÷H) contains W and its three harmonic associates and its equation is :

Naturally, the Lucas cubic K007 is the simplest example of such cubic with W = G, W^2 = G, U = X(69).

When W = K, we obtain K172 with W^2 = X(32) and U = O.

The following table gives a selection of pK(W^2,U) with centers denoted i for X(i). P refers to Clark Kimberling's Z(U,P) cubics.

W	W^2	U	P	cubic	centers on the cubic
1	6	63	1	K343	1, 9, 19, 40, 57, 63, 84, 610, 1712, 2184
2	2	69	31	K007	2, 4, 7, 8, 20, 69, 189, 253, 329, 1032, 1034
4	393	2	255		2, 4, 278, 281, 393, 1249
6	32	3	75	K172	3, 6, 25, 55, 56, 64, 154, 198, 1033, 1035, 1436
7	279	348	1253		2, 7, 278, 279, 347, 348, 1440
8	346	345	1106		2, 8, 280, 281, 345, 346
9	220	78	269
10	594	306	849		10, 226, 306, 1826, 2321
19	2207	1	326		1, 19, 33, 34, 204, 207, 1096, 2331
31	1501	48	561
37	1500	72	757
57	1407	77	200
63	394	326	1096		1, 63, 77, 78, 326
75	76	304	560		75, 85, 92, 304, 309, 312, 322
100	1252	1332	244
513	1015	905	765
523	115	525	1101