[Antreas P. Hatzipolakis]:
Let ABC be a triangle, MaMbMc the medial triangle, P a point and A'B'C' the:
1. pedal
2. cevian
triangle of P
Denote:
M1M2M3 = the medial triangle of A'B'C'
P' = the same to P point of MaMbMc = complement of P
A"B"C" = the pedal triangle of P' wrt triangle MaMbMc
M1M2M3, A"B"C"are perspective.
Which are the perspectors (for pedal, cevian cases) on the line at infinity in terms of P?
(M1A", M2B", M3C" are parallels)
Hi Antreas,
Pedal version:
P{p,q,r}->
a^2 (2 b^2 c^2 p+a^2 c^2 q-b^2 c^2 q-c^4 q+a^2 b^2 r-b^4 r-b^2 c^2 r) : :
Examples:
--------------------------------------------------
P = X(9) ->
X(6)X(57) ∩ X(30)X(511) =
= a*(a^3*b + a^2*b^2 - a*b^3 - b^4 + a^3*c - 4*a^2*b*c + a*b^2*c + 2*b^3*c + a^2*c^2 + a*b*c^2 - 2*b^2*c^2 - a*c^3 + 2*b*c^3 - c^4) : :
= lies on these lines: {1, 24328}, {2, 374}, {6, 57}, {7, 2262}, {9, 18725}, {19, 6180}, {30, 511}, {37, 18161}, {44, 16560}, {65, 4644}, {69, 189}, {77, 198}, {101, 6510}, {141, 3452}, {144, 21871}, {193, 3210}, {241, 2183}, {322, 20348}, {599, 31142}, {651, 2182}, {942, 4667}, {960, 4643}, {999, 1386}, {1100, 18162}, {1108, 1423}, {1122, 4000}, {1229, 20248}, {1350, 6282}, {1351, 2095}, {1436, 7013}, {1443, 11349}, {1486, 30621}, {1604, 7053}, {1814, 32677}, {1829, 23154}, {1901, 5929}, {1905, 24476}, {1944, 3732}, {1992, 2094}, {2093, 3751}, {2096, 6776}, {3056, 17642}, {3057, 4419}, {3242, 7962}, {3416, 3421}, {3589, 6692}, {3698, 4470}, {3740, 17251}, {3763, 20196}, {3812, 4670}, {3820, 3844}, {3882, 25083}, {4259, 7960}, {4363, 5836}, {4454, 14923}, {4641, 26934}, {4659, 10914}, {4662, 4690}, {4708, 24317}, {4748, 25917}, {5011, 10756}, {5060, 16702}, {5085, 21164}, {5480, 7682}, {5908, 6260}, {5909, 6245}, {5942, 21279}, {6244, 12329}, {7011, 34052}, {7202, 8609}, {9432, 26273}, {9943, 24683}, {9954, 10859}, {10387, 10388}, {10391, 17441}, {11677, 30620}, {15587, 21867}, {16284, 20719}, {18675, 28369}, {20080, 20214}, {20262, 21239}, {21370, 34048}, {21785, 28022}, {22129, 24611}, {25274, 30082}
= psi-transform of X(2)
= crossdifference of every pair of points on line {6, 3900}
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 2097, 57}, {651, 7291, 2182}, {2183, 3942, 241}, {17441, 26892, 10391}
--------------------------------------------------
P = X(11)->
X(6)X(906) ∩ X(30)X(511) =
= a^2*(a^5*b^2 - a^4*b^3 - 2*a^3*b^4 + 2*a^2*b^5 + a*b^6 - b^7 - a^4*b^2*c + 2*a^3*b^3*c - 2*a*b^5*c + b^6*c + a^5*c^2 - a^4*b*c^2 - 2*a^3*b^2*c^2 + 2*a^2*b^3*c^2 - a*b^4*c^2 + b^5*c^2 - a^4*c^3 + 2*a^3*b*c^3 + 2*a^2*b^2*c^3 - b^4*c^3 - 2*a^3*c^4 - a*b^2*c^4 - b^3*c^4 + 2*a^2*c^5 - 2*a*b*c^5 + b^2*c^5 + a*c^6 + b*c^6 - c^7) : :
= lies on these lines: {6, 906}, {30, 511}, {52, 3811}, {55, 1331}, {56, 1813}, {1216, 10916}, {2979, 24477}, {3060, 25568}, {3189, 5889}, {3908, 14686}, {5446, 21077}, {12437, 31732}, {16980, 32049}, {24391, 31737}
--------------------------------------------------
P = X(13)->
Q(13) =
X(6)X(2981)∩ X(30)X(511) =
= a^2*(Sqrt[3]*b^2*c^2*(2*a^2 - b^2 - c^2) + 2*(a^2*b^2 - b^4 + a^2*c^2 - c^4)*S) : :
= lies on these lines: {2, 11624}, {6, 2981}, {30, 511}, {51, 33459}, {69, 300}, {141, 16536}, {373, 33474}, {2979, 5859}, {3060, 5858}, {3917, 33458}, {5463, 30439}, {5615, 9145}, {5640, 9761}, {5650, 33475}, {7998, 9763}, {9115, 15544}
= isogonal conjugate of Q*(13)
= {X(11126),X(17403)}-harmonic conjugate of X(19294)
--------------------------------------------------
= {X(11126),X(17403)}-harmonic conjugate of X(19294)
--------------------------------------------------
isog of P = X(13)
Q*(13) =
= ISOGONAL CONJUGATE OF Q(13) =
= 1/((Sqrt[3]*b^2*c^2*(2*a^2 - b^2 - c^2) + 2*(a^2*b^2 - b^4 + a^2*c^2 - c^4)*S)) : :
= lies on the circumcircle and these lines: {2, 10409}, {14, 9202}, {99, 11146}, {110, 396}, {112, 463}, {476, 11141}, {2380, 20579}, {5618, 16463}, {5995, 8014}, {6779, 9203}, {9112, 16806}
= isogonal conjugate of Q(13)
= orthoptic circle of the Steiner inellipse inverse of X(15609)
= X(5472)-cross conjugate of X(11085)
= isogonal conjugate of Q(13)
= orthoptic circle of the Steiner inellipse inverse of X(15609)
= X(5472)-cross conjugate of X(11085)
--------------------------------------------------
P = X(14)->
Q(14) =
X(6)X(6151) ∩ X(30)X(511) =
= a^2*(Sqrt[3]*b^2*c^2*(2*a^2 - b^2 - c^2) - 2*(a^2*b^2 - b^4 + a^2*c^2 - c^4)*S) : :
= lies on these lines: {2, 11626}, {6, 6151}, {30, 511}, {51, 33458}, {69, 301}, {141, 16537}, {373, 33475}, {2979, 5858}, {3060, 5859}, {3917, 33459}, {5464, 30440}, {5611, 9145}, {5640, 9763}, {5650, 33474}, {7998, 9761}, {9117, 15544}
= isogonal conjugate of Q*(14)
= {X(11127),X(17402)}-harmonic conjugate of X(19295)
= {X(11127),X(17402)}-harmonic conjugate of X(19295)
--------------------------------------------------
isog of P = X(14)
= ISOGONAL CONJUGATE OF Q(14) =
= 1/((Sqrt[3]*b^2*c^2*(2*a^2 - b^2 - c^2) + 2*(a^2*b^2 - b^4 + a^2*c^2 - c^4)*S)) : :
= lies on the circumcircle and these lines: {2, 10410}, {13, 9203}, {99, 11145}, {110, 395}, {112, 462}, {476, 11142}, {2381, 20578}, {5619, 16464}, {5994, 8015}, {6780, 9202}, {9113, 16807}
= isogonal conjugate of Q(14)
= orthoptic circle of the Steiner inellipse inverse of X(15610)
= X(5471)-cross conjugate of X(11080)
--------------------------------------------------
= orthoptic circle of the Steiner inellipse inverse of X(15610)
= X(5471)-cross conjugate of X(11080)
--------------------------------------------------
P = X(37)->
X(6)X(63) ∩ X(30)X(511) =
= a*(a^3*b + a^2*b^2 - a*b^3 - b^4 + a^3*c + a*b^2*c + a^2*c^2 + a*b*c^2 - 2*b^2*c^2 - a*c^3 - c^4) : :
= lies on these lines: {6, 63}, {30, 511}, {65, 4363}, {69, 321}, {72, 4259}, {141, 226}, {193, 17147}, {210, 17251}, {320, 17789}, {326, 2178}, {599, 31164}, {942, 4670}, {960, 4364}, {993, 1386}, {1122, 7232}, {1155, 17977}, {1350, 18446}, {1478, 3416}, {1764, 7289}, {1959, 8609}, {2245, 25083}, {2262, 4361}, {3057, 17318}, {3589, 5745}, {3683, 16792}, {3729, 21853}, {3751, 4424}, {3763, 31266}, {3779, 24326}, {3812, 4472}, {3822, 3844}, {3868, 4644}, {3869, 4419}, {3874, 4667}, {3876, 4748}, {3916, 5135}, {3954, 4503}, {4659, 5903}, {4665, 5836}, {4708, 5044}, {4795, 24473}, {4798, 5439}, {5006, 16702}, {5085, 21165}, {5440, 33844}, {7262, 16793}, {10477, 24476}, {10754, 11611}, {12635, 24328}, {17262, 21871}, {18252, 20713}, {18611, 23075}, {18726, 21061}, {20715, 24699}, {22277, 22325}, {24333, 25368}, {24424, 24705}, {24441, 31165}
= crossdifference of every pair of points on line {6, 8678}
--------------------------------------------------
P = X(38)->
X(6)X(3874) ∩ X(30)X(511) =
= a*(a^4*b - b^5 + a^4*c - a^2*b^2*c + 2*a*b^3*c - a^2*b*c^2 - b^3*c^2 + 2*a*b*c^3 - b^2*c^3 - c^5) : :
= lies on these lines: {6, 3874}, {10, 24476}, {30, 511}, {69, 1930}, {141, 3678}, {182, 12005}, {193, 17489}, {611, 18389}, {1386, 3881}, {1428, 5083}, {1469, 15556}, {3242, 3878}, {3313, 23156}, {3618, 18398}, {3663, 4523}, {3751, 3868}, {3811, 7289}, {3844, 4015}, {3869, 16496}, {3873, 16475}, {3889, 16491}, {4973, 5096}, {6583, 18583}, {10516, 15064}, {12432, 24471}, {20455, 20715}, {22769, 22836}, {25050, 32846}, {32118, 32935}
--------------------------------------------------
P = X(42)->
X(10)X(69) ∩ X(30)X(511) =
= 2*a^3 + 3*a^2*b - 2*a*b^2 - b^3 + 3*a^2*c - b^2*c - 2*a*c^2 - b*c^2 - c^3 : :
= lies on these lines: {1, 193}, {6, 1125}, {8, 17116}, {10, 69}, {30, 511}, {42, 4001}, {44, 4966}, {63, 4028}, {141, 3634}, {226, 32853}, {238, 4684}, {239, 24231}, {306, 32912}, {320, 1738}, {355, 11898}, {551, 1992}, {599, 3828}, {611, 13405}, {895, 13605}, {908, 32919}, {940, 4104}, {946, 1351}, {984, 3879}, {991, 3811}, {1054, 5212}, {1266, 4716}, {1326, 6629}, {1350, 12512}, {1352, 19925}, {1353, 1385}, {1386, 3629}, {1468, 4101}, {1469, 4298}, {1698, 3620}, {1742, 6765}, {1757, 3912}, {2321, 32935}, {3011, 16704}, {3056, 12575}, {3242, 3635}, {3244, 11008}, {3416, 3626}, {3555, 21746}, {3576, 14912}, {3589, 19878}, {3616, 17331}, {3618, 19862}, {3630, 4691}, {3631, 3844}, {3679, 11160}, {3685, 20072}, {3686, 24325}, {3687, 32913}, {3696, 17365}, {3717, 32846}, {3729, 4133}, {3755, 4655}, {3763, 31253}, {3775, 5750}, {3790, 17373}, {3817, 14853}, {3826, 17376}, {3836, 4753}, {3886, 24695}, {3914, 32859}, {3920, 20086}, {3932, 17374}, {3977, 4062}, {3980, 4061}, {4026, 17344}, {4035, 4438}, {4078, 4851}, {4138, 33137}, {4260, 12436}, {4297, 6776}, {4353, 4856}, {4357, 4649}, {4385, 34282}, {4429, 17361}, {4700, 4974}, {4722, 5294}, {4745, 15533}, {4780, 24248}, {4847, 32946}, {4899, 32847}, {4938, 32848}, {5032, 25055}, {5050, 10165}, {5052, 12263}, {5093, 5886}, {5095, 11720}, {5249, 32864}, {5477, 11711}, {5480, 12571}, {5542, 16825}, {5691, 5921}, {5713, 10916}, {5788, 21077}, {5905, 17156}, {6210, 6762}, {9798, 19588}, {9967, 31738}, {10164, 10519}, {10171, 14561}, {10477, 12572}, {10753, 21636}, {10754, 11599}, {10755, 21630}, {10759, 21635}, {10761, 11814}, {12513, 31394}, {13211, 32244}, {15481, 17243}, {15569, 17332}, {16830, 20090}, {17023, 28650}, {17348, 25557}, {17353, 33087}, {17363, 24349}, {17781, 32915}, {18440, 31673}, {18483, 21850}, {19868, 33682}, {21060, 29649}, {24210, 33066}, {25006, 32949}, {26015, 32843}, {26227, 31303}, {26723, 33069}, {29639, 31034}, {30768, 31017}, {31730, 33878}
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {69, 3751, 10}, {4722, 33081, 5294}, {4851, 5220, 4078}
--------------------------------------------------
P = X(51)->
X(6)X(140) ∩ X(30)X(511) =
= 2*a^6 - 7*a^4*b^2 + 6*a^2*b^4 - b^6 - 7*a^4*c^2 + b^4*c^2 + 6*a^2*c^4 + b^2*c^4 - c^6 : :
= lies on these lines: {2, 5093}, {3, 193}, {4, 11898}, {5, 69}, {6, 140}, {30, 511}, {49, 19121}, {51, 10128}, {52, 9825}, {66, 18356}, {141, 576}, {159, 9925}, {182, 3530}, {195, 34002}, {230, 1570}, {265, 32244}, {323, 468}, {325, 10011}, {340, 6530}, {381, 11160}, {382, 5921}, {394, 6677}, {487, 12314}, {488, 12313}, {546, 1352}, {547, 599}, {548, 1350}, {549, 1992}, {550, 6776}, {575, 12108}, {597, 10124}, {613, 28369}, {632, 3618}, {895, 10264}, {1147, 19154}, {1368, 6515}, {1484, 10755}, {1511, 5095}, {1513, 7779}, {1595, 12167}, {1656, 3620}, {1843, 10263}, {1993, 6676}, {1994, 7499}, {2080, 6390}, {2979, 10691}, {3056, 15172}, {3095, 7767}, {3098, 8550}, {3167, 10154}, {3292, 32269}, {3313, 15074}, {3524, 33748}, {3580, 5159}, {3589, 5097}, {3627, 18440}, {3630, 3850}, {3631, 24206}, {3751, 5690}, {3785, 10983}, {3818, 3861}, {3853, 15069}, {3917, 7734}, {4220, 20086}, {5028, 5305}, {5032, 5054}, {5066, 10516}, {5085, 12100}, {5092, 12007}, {5111, 15993}, {5181, 10272}, {5188, 7890}, {5446, 14913}, {5476, 10109}, {5477, 33813}, {5562, 13142}, {5609, 32114}, {5656, 12164}, {5774, 15973}, {5876, 12294}, {5889, 31829}, {5943, 13361}, {6101, 9967}, {6194, 7837}, {6243, 6403}, {6329, 22330}, {6391, 11411}, {6467, 10625}, {6593, 13392}, {6661, 22521}, {6675, 15988}, {6823, 12160}, {6998, 20090}, {7380, 17343}, {7387, 19588}, {7495, 11004}, {7575, 32220}, {7762, 12251}, {7788, 9753}, {8359, 32447}, {8584, 11812}, {8703, 25406}, {9300, 15819}, {9822, 10095}, {9924, 9936}, {9969, 23411}, {9974, 13925}, {9975, 13993}, {10112, 12024}, {10113, 32275}, {10168, 20583}, {10300, 18911}, {10627, 11574}, {10733, 32272}, {10759, 11698}, {11178, 11737}, {11179, 31884}, {11180, 15687}, {11255, 12359}, {11412, 12022}, {11694, 15303}, {12017, 15712}, {12107, 15577}, {12121, 32234}, {12161, 16197}, {12272, 16658}, {12322, 12602}, {12323, 12601}, {12325, 15559}, {12584, 25329}, {12811, 19130}, {13331, 22677}, {13340, 15531}, {13346, 23328}, {13383, 19139}, {13451, 29959}, {13488, 18436}, {13562, 14449}, {14848, 15699}, {14891, 17508}, {15073, 26926}, {15122, 19348}, {15462, 16531}, {15557, 25043}, {15812, 18952}, {16238, 20806}, {16789, 25337}, {18934, 23335}, {19126, 32046}, {19128, 22115}, {19129, 34148}, {19697, 32134}, {20304, 32257}, {25321, 32609}, {25338, 32113}, {32448, 32451}
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 193, 1353}, {4, 20080, 11898}, {69, 1351, 5}, {141, 576, 18583}, {141, 18583, 3628}, {599, 5102, 14561}, {1352, 11477, 21850}, {1352, 21850, 546}, {1992, 10519, 5050}, {2979, 11245, 10691}, {3527, 11487, 5}, {5050, 10519, 549}, {5480, 18358, 3850}, {6776, 33878, 550}, {14848, 21356, 15699}
--------------------------------------------------
P = X(63)->
ISOGONAL CONJUGATE OF (15344) =
= a*(a^2 - b^2 - c^2)*(a^2*b + b^3 + a^2*c - 2*a*b*c - b^2*c - b*c^2 + c^3) : :
= lies on these lines: {1, 7083}, {3, 7289}, {6, 169}, {10, 25365}, {30, 511}, {63, 17441}, {65, 3751}, {68, 12587}, {69, 72}, {105, 15382}, {116, 31897}, {141, 5044}, {182, 9940}, {193, 1829}, {228, 18607}, {238, 20601}, {242, 3732}, {354, 3167}, {651, 1876}, {1069, 12595}, {1071, 6776}, {1147, 13373}, {1214, 20760}, {1282, 5018}, {1350, 31793}, {1351, 24474}, {1352, 5777}, {1353, 24475}, {1385, 22769}, {1386, 5045}, {1428, 3660}, {1439, 23603}, {1736, 21362}, {1738, 20455}, {1814, 7193}, {1818, 3942}, {1824, 5905}, {1828, 12649}, {1843, 14054}, {1902, 5921}, {1992, 24473}, {2262, 5781}, {3033, 9436}, {3057, 16496}, {3173, 5173}, {3242, 9957}, {3579, 12329}, {3618, 5439}, {3620, 3876}, {3927, 4047}, {4259, 7352}, {4260, 10481}, {4298, 13572}, {4463, 32859}, {4523, 4655}, {4663, 31794}, {5050, 10202}, {5085, 11227}, {5096, 5122}, {5138, 11018}, {5480, 5806}, {5504, 10100}, {5776, 10441}, {5880, 21867}, {5885, 8548}, {6147, 9895}, {6467, 23154}, {6510, 17976}, {6583, 19139}, {6708, 20256}, {9925, 15178}, {9928, 34339}, {10157, 10516}, {10167, 25406}, {10477, 30625}, {11573, 11574}, {12429, 14872}, {12586, 31937}, {12723, 24695}, {13605, 23296}, {14913, 29957}, {15076, 17635}, {16465, 26892}, {16491, 17609}, {16560, 23693}, {17102, 20805}, {17615, 22321}, {17975, 22148}, {18651, 21015}, {18734, 23167}, {20078, 20243}, {20254, 22149}, {20752, 20811}, {21167, 33575}, {32126, 32263}
= isogonal conjugate of X(15344)
= isotomic conjugate of the polar conjugate of X(3290)
= crossdifference of every pair of points on line {6, 15313}
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 24476, 942}, {4523, 4655, 18252}
--------------------------------------------------
P = X(68)->
X(6)X(1147) ∩ X(30)X(511) =
= a^2*(a^2 - b^2 - c^2)*(a^6*b^2 - a^4*b^4 - a^2*b^6 + b^8 + a^6*c^2 - 4*a^4*b^2*c^2 + 3*a^2*b^4*c^2 - 4*b^6*c^2 - a^4*c^4 + 3*a^2*b^2*c^4 + 6*b^4*c^4 - a^2*c^6 - 4*b^2*c^6 + c^8) : :
= lies on these lines: {3, 6391}, {4, 12271}, {5, 14913}, {6, 1147}, {20, 12282}, {30, 511}, {49, 21637}, {51, 3167}, {52, 193}, {67, 19477}, {68, 69}, {141, 5449}, {155, 1351}, {159, 32048}, {182, 8548}, {265, 32260}, {389, 1353}, {487, 12604}, {488, 12603}, {575, 22829}, {576, 9925}, {895, 5504}, {1112, 21847}, {1350, 7689}, {1352, 9927}, {1469, 19471}, {1495, 12310}, {1692, 32661}, {1993, 27365}, {2931, 32127}, {3056, 9931}, {3060, 7714}, {3095, 19597}, {3098, 9938}, {3292, 18449}, {3448, 32249}, {3580, 32263}, {3629, 21852}, {3751, 9928}, {3779, 12417}, {3818, 22661}, {5028, 23128}, {5050, 5892}, {5095, 11557}, {5102, 9971}, {5447, 11574}, {5448, 5480}, {5562, 11898}, {5596, 12420}, {5654, 11188}, {5921, 12162}, {6102, 21851}, {6239, 12222}, {6291, 12602}, {6400, 12221}, {6406, 12601}, {6504, 14593}, {6776, 12118}, {6803, 17040}, {7387, 9924}, {8538, 20806}, {9544, 19123}, {9730, 14912}, {9820, 9822}, {9973, 11477}, {10111, 32285}, {10116, 12421}, {10170, 14852}, {10282, 19154}, {10625, 11411}, {11412, 20080}, {11562, 32234}, {11579, 12901}, {12163, 33878}, {12166, 12167}, {12223, 12510}, {12224, 12509}, {12293, 12294}, {12383, 32248}, {12412, 32276}, {12419, 32264}, {12584, 19138}, {12590, 19486}, {12591, 19487}, {13137, 17932}, {13367, 19129}, {13383, 15585}, {14561, 29959}, {15045, 33748}, {15118, 19509}, {15123, 23296}, {15583, 23335}, {18475, 19131}, {18934, 18935}, {19196, 19197}, {19458, 19459}, {20302, 24206}, {20794, 30258}, {21639, 22115}, {21650, 32272}, {21850, 22660}, {32114, 32123}, {32191, 32455}
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {{6, 9937, 19141}, {69, 9967, 1216}, {69, 15073, 9967}, {193, 6403, 52}, {1147, 12235, 5462}, {1351, 1843, 5446}, {1351, 19588, 155}, {3167, 14914, 5093}, {9926, 19141, 6}, {9937, 15316, 1147}, {10607, 10608, 3}, {11898, 18438, 5562}
--------------------------------------------------
P = X(76)->
X(6)X(694) ∩ X(30)X(511) =
= a^2*(a^4*b^4 - a^2*b^6 - a^2*b^4*c^2 + a^4*c^4 - a^2*b^2*c^4 + 2*b^4*c^4 - a^2*c^6) : :
= lies on these lines: {2, 6784}, {6, 694}, {30, 511}, {69, 290}, {76, 4173}, {141, 7668}, {182, 9145}, {193, 8264}, {211, 7805}, {263, 1992}, {373, 12093}, {385, 11673}, {597, 34236}, {671, 6787}, {805, 14931}, {895, 9513}, {1355, 9413}, {1356, 7170}, {1576, 3506}, {1843, 27377}, {1976, 4558}, {1987, 6391}, {2421, 9149}, {2482, 3111}, {3056, 24437}, {3060, 7837}, {3098, 9142}, {3491, 5254}, {3571, 7063}, {3618, 31639}, {3629, 25326}, {4590, 34238}, {5943, 9300}, {5989, 17970}, {6054, 6785}, {6071, 12833}, {6072, 13137}, {6128, 29959}, {7062, 9414}, {7760, 27374}, {7813, 14962}, {7838, 27375}, {7840, 13207}, {8598, 32442}, {10602, 16098}, {10765, 14948}, {11184, 13240}, {12157, 18823}, {14609, 30495}, {14981, 31850}, {15630, 15631}, {15991, 19581}, {21320, 28369}, {24206, 33548}, {30534, 30535}
= isotomic conjugate of the isogonal conjugate of X(21444)
= crossdifference of every pair of points on line {6, 804}
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 694, 1084}, {69, 25051, 20021}, {69, 25332, 670}, {141, 25324, 7668}, {6784, 6786, 2}, {7840, 13207, 33873}, {15630, 15631, 22103}
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Best regards,
Peter Moses.
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