Let pK(Ω = p : q : r, P = u : v : w) be a pivotal cubic with isopivot P*, the barycentric quotient Ω ÷ P. We suppose Ω ≠ P^2 in order to discard pKs decomposed into the cevian lines of P. This cubic is anharmonically equivalent to K131 = pK(X31, X171) if and only if : ∑ a (b-c) (a^2-b c) (b^2 + b c + c^2) q r u^2 = 0. (E) (E) can be construed as follows : • X2, X292, P^2 ÷ Ω are collinear, • Ω, Ω x X292, P^2 are collinear, • P, P*, P* x X292 are collinear, X x Y denotes the barycentric product of X and Y. For a given pole Ω, (E) shows that P must lie on a diagonal conic and for a given pivot P, (E) shows that Ω must lie on a circum-conic. *** The following table shows a large selection of cubics equivalent to K131. See the related Table 66 and also Table 68.
 cubic Ω P n X(i) on the cubic for i < 3249 note K131 X31 X171 1 2, 31, 42, 43, 55, 57, 81, 171, 365, 846, 893, 2162, 2248 (6) K132 X1 X894 0 6, 7, 9, 37, 75, 86, 87, 192, 256, 366, 894, 1045, 1654 (6) K135 X1911 X291 1 1, 6, 42, 57, 239, 291, 292, 672, 894, 1757, 1967 (4) K136 X292 X292 1 1, 2, 37, 87, 171, 238, 241, 291, 292, 1575, 1581, 2664 (3) K155 X31 X238 1 1, 2, 6, 31, 105, 238, 292, 365, 672, 1423, 1931, 2053, 2054, 2106, etc (1) K251 X238 X2 0 1, 2, 9, 86, 238, 239, 292, 673, 893, 1447, 1929, 1966, 2238 (2) K323 X1 X239 0 1, 2, 6, 75, 239, 291, 366, 518, 673, 1575, 2319, 2669, 3212, 3226 (1) K673 X1914 X6 1 1, 6, 43, 81, 238, 239, 256, 291, 294, 1580, 2068, 2069, 2238, 2665 (5) K744 X75 X1909 -1 1, 8, 10, 76, 85, 257, 274, 330, 1655, 1909 (6) K766 X75 X350 -1 1, 2, 75, 76, 335, 350, 726, 2481 (1) K767 X350 X75 -1 2, 8, 75, 239, 256, 274, 291, 350, 740, 1281, 2481 (2) K768 X335 X291 0 2, 10, 75, 239, 291, 330, 335, 726, 894, 1916 (3) K769 X291 X335 0 1, 2, 7, 37, 291, 335, 350, 518, 694, 1909 (4) K770 X239 X1 0 1, 2, 86, 192, 239, 257, 335, 350, 385, 740, 3226 (5) K771 X560 X1914 2 1, 6, 31, 32, 727, 1326, 1403, 1438, 1911, 1914, 2223 (1) K772 X14598 X292 2 6, 31, 56, 171, 213, 238, 292, 741, 1911, 2223 (4) K773 X14599 X31 2 6, 31, 58, 238, 292, 727, 893, 1691, 1914, 2176, 2195 (5) K774 X2210 X1 1 1, 6, 55, 81, 105, 238, 385, 904, 1429, 1911, 1914 (2) K775 X1922 X1911 2 1, 6, 42, 172, 292, 694, 741, 1458, 1911, 1914, 2162, 3009 (3) K868 X334 X334 -1 2, 10, 75, 85, 334, 335, 1581, 1920, 1921 (4) K960 X1914 X8424 21, 256, 846, 1281, 1284, 1580 K961 X1914 X8301 105, 291, 1281, 1282, 1929, 2108 K986 X18891 X76 -1 75, 76, 257, 310, 312, 335, 350, 1921, 1926 (2) K987 X18892 X6 2 6, 31, 41, 58, 1428, 1438, 1580, 1914, 1922, 2210 (2) K988 X18893 X1911 3 31, 32, 172, 604, 1911, 1914, 1918, 1922, 1927 (4) K993 X18894 X32 3 31, 32, 904, 1333, 1911, 1914, 1933, 2209, 2210 (5) K994 X18895 X335 -1 75, 76, 321, 334, 335, 350, 1909, 1934 (3) K996 X1921 X2 -1 2, 75, 274, 334, 350, 1921, 1966 (5) K997 X18897 X1922 3 6, 31, 213, 1911, 1922, 1967, 2210 (3) K1002 X2 X4645 0 2, 7, 8, 1654, 2113 (7) K1003 X32 X17798 2 6, 55, 56, 2248 (7) K1006 X560 X172 2 1, 32, 41, 56, 58, 172, 213, 904, 2176 (6) K1007 X561 X1920 -2 2, 310, 312, 321, 561, 1920 (6) K1017 X869 X1 1, 2, 6, 55, 192, 869, 984, 1002, 2276 K1018 X985 X14621 1, 2, 6, 7, 870, 985, 1001, 2162, 2344 K1019 X18900 X6 1, 6, 31, 41, 43, 869, 1469, 2276, 2279 K1020 X561 X1921 -2 2, 75, 76, 334, 561, 1921 (1) K1021 X1917 X2210 3 6, 31, 32, 560, 1922, 2210 (1) K1025 X6 X3509 1 1, 9, 57, 846, 1282, 1757, 1929 (7) K1026 X6 X3510 1, 43, 87, 1045, 2664, 2665 K1038 X984 X2 1, 2, 9, 75, 984, 2276 K1040 X16514 X1 1, 238, 291, 740, 984, 3736, 3783, 3795, 3802, 7220, 7281, 8298, 17793 K1041 X2276 X518 1, 8, 291, 518, 984, 1002, 1282, 1469, 2113, 3783, 3789, 17794
 Notes : each number refers to cubics with equations of the same type and with the same color in the table. Furthermore, Kxxxx(n+1) = X(1) x Kxxxx(n). Note that X(335) is the isotomic conjugate of X(239). (1) : K323(n) = pK(X1^(2n+1), X1^n x X239) (2) : K251(n) = pK(X1^(2n+1) x X239, X1^n) (3) : K768(n) = pK(X1^(2n) x X335, X1^(n+1) x X335) (4) : K769(n) = pK(X1^(2n+1) x X335, X1^n x X335) (5) : K770(n) = pK(X1^(2n) x X239, X1^(n+1)) (6) : K132(n) = pK(X1^(2n+1), X1^n x X894) (7) : K1002(n) = pK(X1^(2n), X1^n x X4645) Remarks : • these equations clearly show that the cubics above are weak for any n. It follows that the symbolic substitution SS{a -> a^2} transforms each cubic into a strong pK equivalent to K020 as in Table 66. • when Ω = X2 (resp. P = X2), the complement (resp. anticomplement) of pK(Ω, P) is another pK equivalent to K131. *** Additional data by Peter Moses The following table shows other cubics passing through at least 10 ETC centers. All of them are simple barycentric products, see column 3.
 Ω, P X(i) on the cubic pK(Ω, P) for i <18859 products 200, 3685 8, 9, 55, 192, 312, 3685, 3693, 4182, 4876, 14942 X(8) x K323 269, 7176 1, 56, 65, 85, 279, 1432, 1434, 3212, 7153, 7176, 17084 X(7) x K132 341, 17787 9, 75, 314, 346, 2321, 3596, 4110, 4451, 7155, 17787 X(8) x K744 756, 740 10, 37, 42, 321, 740, 3930, 4179, 6542, 13576, 17759 X(10) x K323 756, 1215 2, 42, 210, 226, 321, 756, 1215, 3971, 4179, 16606 X(10) x K132 765, 3570 100, 101, 190, 660, 666, 668, 1026, 3570, 8709, 17934 X(668) x K155 765, 18047 99, 101, 644, 664, 668, 932, 1018, 3903, 4595, 18047 X(668) x K131 1253, 2329 1, 8, 21, 41, 220, 1334, 2053, 2329, 3208, 4166, 8931 X(9) x K132 1253, 3684 8, 9, 41, 43, 55, 294, 1282, 2340, 3684, 4166, 7077, 8851 X(9) x K323 3248, 659 513, 514, 649, 659, 665, 667, 1027, 3572, 6373, 18001 X(513) x K323 4037, 37 2, 10, 37, 740, 1655, 3948, 3971, 4037, 4039, 4368 X(10) x K770 8300, 385 350, 385, 1447, 1914, 2238, 3509, 3510, 3684, 8844, 18786 X(239) x K132 8300, 4366 1, 238, 239, 350, 1914, 3253, 4366, 6654, 8299, 17475 X(239) x K323