Symbolic substitutions are introduced in C. Kimberling, "Symbolic substitutions in the transfigured plane of a triangle," Aequationes Mathematicae 73 (2007) 156-171. See also Manfred Evers, Symbolic substitution has a geometric meaning, Forum Geometricorum, vol.14 (2014), pp 217--232. *** A symbolic substitution of the form SS{a → a'} can be defined in different equivalent ways. Indeed, a' can be a function of the lengths a, b, c of the sides, or a (most of the time trigonometric) function of the angles A, B, C, or the first barycentric coordinate of a triangle center X(n) in which case we shall write SS{Xn} in short. For instance, SS{X3} is SS{a → a^2(-a^2 + b^2 + c^2)} or SS{a → a^2 SA} or SS{a → sin 2A}. As mentioned in ETC, these symbolic substitutions map lines to lines, conics to conics, cubics to cubics, etc. In this page, we will not examine their properties and will focus exclusively on some remarkable cubics with important contributions by Peter Moses. N is the number of centers of the SS transform of the cubic Knnn. Two groups of cubics are particularly involved, those mentioned in pages K323 and K718.
 N Knnn SS X(i) on the SS transform of Knnn for i in ... 14 K1083 SS{X9} {1, 2, 8, 9, 144, 165, 210, 518, 3057, 3307, 3308, 15587, 24646, 24647} K1084 14 K907 SS{X188} 14 K1083 SS{X3} {2, 3, 6, 20, 69, 154, 511, 2574, 2575, 3917, 5907, 6467, 13414, 13415} K1085 12 K779 SS{X366} {2, 7, 8, 346, 350, 3912, 4876, 5853, 14942, 14943, 18025, 20533} pK(X8, X3912) 12 K323 SS{X8} 12 K766 SS{X7} 12 K738 SS{X508} 12 K623 SS{X9} 12 K780 SS{X188} 12 K778 SS{X366} {2, 8, 75, 144, 518, 673, 3912, 4437, 9311, 9436, 18025, 20935} pK(X8, X3912) 12 K770 SS{X8} 12 K767 SS{X7} 12 K356 SS{X508} 12 K718 SS{X188} 12 K777 SS{X366} {1, 2, 7, 516, 673, 2481, 3729, 3912, 6185, 10025, 10405, 14942} pK(X673, X14942) 12 K768 SS{X8} 12 K769 SS{X7} 12 K354 SS{X508} 12 K776 SS{X188} 12 K776 SS{X366} {2, 8, 9, 239, 673, 2319, 2481, 4373, 5853, 9312, 9436, 14942} pK(X14942, X673) 12 K769 SS{X8} 12 K768 SS{X7} 12 K322 SS{X508} 12 K777 SS{X188} 12 K1017 SS{X4} {2, 4, 193, 263, 393, 3424, 6525, 6620, 6776, 7735, 9292, 9752} pK(X6620, X4) 12 K1016 SS{Sqrt(X4)} 12 K982 SS{X9} {1, 2, 9, 192, 518, 1280, 2340, 3158, 3693, 4876, 6168, 8299} pK(X2340, X1) 12 K783 SS{X188} 12 K506 SS{X4} {2, 20, 64, 235, 253, 393, 1105, 6330, 6525, 6526, 14249, 16318} 11 K718 SS{X366} {2, 7, 85, 145, 335, 518, 3912, 9436, 10029, 14942, 16593} pK(X9436, X7) 11 K767 SS{X8} 11 K770 SS{X7} 11 K739 SS{X508} 11 K778 SS{X188} 11 K768 SS{X10} {2, 86, 350, 1125, 6542, 6625, 6650, 9505, 11599, 17770, 20536} pK(X6650, X11599) 11 K623 SS{X239} 11 K769 SS{X86} 11 K251 SS{X8} {2, 8, 75, 3161, 3693, 3717, 3912, 4518, 6376, 6559, 17755} pK(3717, X2) 11 K996 SS{X7} 11 K984 SS{X10} {10, 37, 42, 321, 740, 3930, 4179, 6542, 13576, 17759, 20694} pK(X756, X740) 11 K984 SS{X37} 11 K769 SS{X10} {2, 10, 75, 740, 1213, 6650, 9278, 11599, 17731, 20016, 24731} pK(X11599, X6650) 11 K768 SS{X86} 11 K766 SS{X10} {2, 10, 86, 239, 1509, 6650, 13174, 13610, 17731, 17770, 18827} pK(X6542, X10) 11 K323 SS{X86} 11 K1009 SS{X366} {2, 43, 55, 57, 165, 200, 1376, 2319, 3158, 8056, 19605} pK(X55, X1376) 11 K1009 SS{X188} 11 K357 SS{X366} {1, 2, 9, 291, 518, 2481, 3912, 6184, 9436, 14943, 17755} pK(X518, X2) 11 K357 SS{X188} 11 K294 SS{X37} 10, 37, 42, 213, 1757, 2238, 2664, 18785, 18793, 20683, 21830} 11 K982 SS{X3} {2, 3, 6, 194, 511, 2967, 2987, 3167, 3289, 15143, 17974} 11 K453 SS{X3} {2, 3, 6, 30, 265, 1511, 3163, 3284, 11064, 14910, 14919} 11 K1065 SS{X188} {1, 8, 34, 65, 85, 279, 1212, 2082, 5998, 14584, 21132} 11 K803 SS{X188} {519, 1001, 1320, 3307, 3308, 5239, 5240, 7026, 7043, 10707, 14942} 11 K505 SS{X188} {2, 9, 519, 908, 1000, 1320, 3872, 5239, 5240, 7026, 7043} 11 K364 SS{X188} {4, 10, 21, 40, 78, 145, 188, 280, 3152, 3680, 13583} 11 K506 SS{X9} {2, 7, 8, 55, 220, 480, 1223, 2340, 2346, 3059, 14942} 10 K1010 SS{X366} {7, 9, 75, 144, 192, 346, 3161, 3729, 4373, 7155} pK(X8, X3729) 10 K132 SS{X8} 10 K744 SS{X7} 10 K743 SS{X508} 10 K1011 SS{X9} 10 K675 SS{X188} 10 K251 SS{X4} {2, 4, 132, 232, 264, 297, 1249, 6330, 6530, 6531} pK(X6530, X2) 10 K252 SS{Sqrt(X4)} 10 K996 SS{X69} 10 K136 SS{X4} {2, 4, 6, 98, 230, 419, 3224, 6530, 6531, 16081} pK(X6531, X6531) 10 K787 SS{Sqrt(X4)} 10 K868 SS{X69} 10 K135 SS{X4} {4, 25, 98, 297, 393, 459, 6531, 9308, 16081, 16318} pK(?, X98) 10 K532 SS{Sqrt(X4)} 10 K994 SS{X69} 10 K534 SS{X366} {1, 8, 188, 519, 1320, 5239, 5240, 5541, 7026, 7043} pK(X9, X519) 10 K1080 SS{X9} 10 K001 SS{X188} 10 K996 SS{X4} {2, 69, 76, 325, 5976, 6337, 6374, 6393, 6394, 8781} pK(X6393, X2) 10 K251 SS{X69} 10 K343 SS{X4} {4, 196, 459, 1249, 3176, 3183, 7003, 7149, 8894, 14361} pK(X393, X14361) 10 K174 SS{Sqrt(X4)} 10 K1017 SS{X7} {2, 7, 145, 279, 390, 1002, 3598, 5222, 9309, 9533} pK(X3598, X7) 10 K1016 SS{X508} 10 K253 SS{X8} {1, 2, 9, 346, 3452, 3680, 3752, 6552, 6736, 12640} pK(?, X2) 10 K924 SS{X188} 10 K323 SS{X10} {2, 10, 86, 335, 594, 740, 6542, 10026, 11599, 17762} pK(X10, X6542) 10 K766 SS{X86} 10 K1013 SS{X18297} {2, 75, 76, 330, 870, 871, 4441, 6063, 10009, 20917} pK(X871, ?) 10 K1018 SS{X75} 10 K1014 SS{X366} {1, 2, 75, 85, 87, 870, 4384, 8033, 14621, 24349} pK(X870, X870) 10 K1014 SS{X18297} 10 K699 SS{X366} {335, 673, 3512, 4366, 6650, 6651, 17738, 17755, 18037, 20533} pK(239, X17738) 10 K132 SS{X239} 10 K961 SS{X239} {350, 385, 1447, 1914, 2238, 3509, 3510, 3684, 8844, 18786} 10 K673 SS{X239} {1, 238, 239, 350, 1914, 3253, 4366, 6654, 8299, 17475} 10 K770 SS{X894} {2, 6, 291, 894, 1220, 1281, 3509, 7166, 8424, 17789} 10 K323 SS{X894} {2, 257, 335, 385, 894, 3497, 3509, 6645, 17738, 17739} 10 K982 SS{X37} {2, 10, 37, 210, 740, 1654, 2238, 9278, 13576, 16609} 10 K1079 SS{X3} {3, 6, 219, 222, 394, 1073, 1433, 1498, 7078, 15905} 10 K769 SS{X3} {2, 3, 69, 216, 511, 1972, 1987, 9291, 14941, 16089} 10 K510 SS{X3} {74, 113, 265, 399, 1511, 2931, 5504, 12383, 16163, 20123} 10 K506 SS{X3} {2, 4, 54, 69, 184, 287, 577, 1092, 3289, 5562} 10 K382 SS{X3} {2, 3, 68, 69, 577, 1147, 3289, 3917, 9967, 23195} 10 K332 SS{X3} {6, 69, 371, 372, 5374, 6515, 9937, 11090, 11091, 15316} 10 K323 SS{X3} {2, 3, 264, 287, 401, 511, 577, 1988, 5374, 14941} 10 K254 SS{X3} {2, 3, 68, 97, 264, 317, 324, 5562, 5889, 8795} 10 K168 SS{X3} {2, 317, 371, 372, 577, 1147, 5408, 5409, 10960, 10962} 10 K058 SS{X3} {3, 5, 30, 265, 1807, 7100, 10217, 10218, 15392, 20123} 10 K482 SS{X4} {4, 107, 112, 648, 1294, 2479, 2480, 3163, 6531, 16080} 10 K885 SS{X188} {2, 10, 80, 149, 519, 5239, 5240, 5692, 7026, 7043} 10 K884 SS{X188} {2, 8, 11, 392, 519, 1145, 5239, 5240, 7026, 7043} 10 K696 SS{X8} {1, 145, 346, 2136, 3161, 3680, 4373, 4936, 6736, 23617} 10 K982 SS{X10} {2, 10, 37, 740, 1655, 3948, 3971, 4037, 4039, 4368} 10 K980 SS{X10} {10, 37, 226, 335, 756, 3930, 3948, 3963, 6541, 20715} 10 K506 SS{X10} {1, 2, 75, 335, 594, 756, 1089, 1224, 4037, 4647} 10 K1013 SS{X75} {2, 76, 871, 1502, 2998, 3114, 7034, 10010, 18022, 20023} 10 K775 SS{X75} {75, 76, 321, 334, 1916, 3263, 6063, 17789, 18891, 18895} 10 K423 SS{X75} {2, 75, 76, 141, 183, 308, 327, 10010, 20917, 24273} 10 K252 SS{X75} {2, 76, 308, 1920, 1921, 3978, 6374, 14603, 18277, 18896} 10 K778 SS{X18297} {1, 2, 192, 726, 1575, 3551, 4876, 16557, 20671, 21219} 10 K484 SS{X366} {1, 80, 88, 1015, 2802, 3227, 4370, 4440, 4876, 9278} 10 K482 SS{X366} {1, 80, 88, 100, 190, 292, 644, 1018, 1120, 4370} 10 K273 SS{X366} {1, 2, 7, 88, 519, 679, 903, 1320, 3218, 22464} 10 K239 SS{X366} {2, 514, 519, 900, 903, 1086, 1647, 6548, 6549, 14078}