X662

TRILINEAR POLE OF LINE X1X_21

Let A2B2C2 and A3B3C3 be the 2nd and 3rd Parry triangles. Let A' be the trilinear product A2*A3, and define B' and C' cyclically. The lines AA', BB', CC' concur in X662. (Randy Hutson, February 10, 2016)
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 48, 75 }: Central Line L(48, 75) { 81, 88 }: Central Line L(81, 88) { 99, 101 }: Central Line L(99, 101) { 100, 110 }: Central Line L(100, 110) { 86, 142 }: Central Line L(86, 142) { 60, 404 }: Central Line L(60, 404) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_48 = CROSSPOINT OF X(1) AND X(63) (0.5412, 0.1246) X_75 = ISOTOMIC CONJUGATE OF INCENTER (0.4988, 0.1682) X_81 = CEVAPOINT OF INCENTER AND SYMMEDIAN POINT (0.5849, 0.2510) X_88 = ISOGONAL CONJUGATE OF X(44) (1.0000, 0.0000) X_99 = STEINER POINT (0.1109, 0.4322) X_101 = Ψ(INCENTER, SYMMEDIAN POINT) (0.2143, 0.5249) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.1429, 0.4666) X_110 = FOCUS OF KIEPERT PARABOLA (0.2516, 0.5478) X_86 = CEVAPOINT OF INCENTER AND CENTROID (0.5534, 0.2171) X_142 = COMPLEMENT OF X(9) (0.5506, 0.2192) X_60 = ISOGONAL CONJUGATE OF X(12) (0.5790, 0.2258) X_404 = {X(2),X(3)}-HARMONIC CONJUGATE OF X(21) (0.5538, 0.2429) X_662 = TRILINEAR POLE OF LINE X(1)X(21) (0.1811, 0.4951) A B C X48 X75 X81 X88 X99 X101 X100 X110 X86 X142 X60 X404 X662 (a, b, c) = (5, 6, 7) | AB = 1

X662 (TRILINEAR POLE OF LINE X1X_21) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(1/(\cot B - \cot C) : 1/(\cot C - \cot A) : 1/(\cot A - \cot B)\)
Barycentrics: \((\sin A)/(\cot B - \cot C) : (\sin B)/(\cot C - \cot A) : (\sin C)/(\cot A - \cot B)\)
Search Value (6, 9, 13): \(1.858405610836\)
▸ Alternate Coordinates5 forms

  • Trilinears: \(1/(b^{2} - c^{2}) : :\)

  • Trilinears: \(1/(\cos 2B - \cos 2C) : :\)

  • Trilinears: \(d(a,b,c) : :\), where d(a,b,c) = directed distance of A to line X115X125

  • Barycentrics: \(\cos(B - C) - \cos(C - A) \cos(A - B) : :\)

  • Barycentrics: \(\sec(B - C) - \sec(C - A) \sec(A - B) : :\)

▸ Linear Combinations1 combination

  • X662 = \(6\) X2 - \(5\) X31278, \(3\) X2 + X31297, \(3\) X2 - \(4\) X40539, \(4\) X8287 - \(3\) X31175, \(4\) X8287 - \(5\) X31278, \(2\) X8287 + X31297, \(2\) X21221 - \(3\) X31175, \(2\) X21221 - \(5\) X31278, X21221 - \(4\) X40539, \(3\) X31175 - \(5\) X31278, \(3\) X31175 + \(2\) X31297, \(3\) X31175 - \(8\) X40539, \(5\) X31278 + \(2\) X31297, \(5\) X31278 - \(8\) X40539, X31297 + \(4\) X40539

X662 lies on the cubic K1004, the curve CC9.

▸ Incident Central Lines303 lines

{1, 897}, {2, 2185}, {3, 1098}, {6, 757}, {7, 40622}, {9, 39256}, {10, 501}, {11, 19642}, {19, 8771}, {21, 1156}, {27, 913}, {28, 14868}, {29, 1800}, {31, 36289}, {37, 34053}, {41, 3758}, {43, 6043}, {44, 1931}, {48, 75}, {58, 16477}, {60, 404}, {63, 2159}, {65, 35991}, {69, 24580}, {76, 30882}, {81, 88}, {82, 1178}, {86, 142}, {99, 101}, {100, 110}, {109, 931}, {112, 1310}, {115, 24957}, {141, 24614}, {162, 2617}, {163, 2642}, {214, 759}, {224, 13739}, {229, 34772}, {238, 1326}, {239, 7113}, {243, 425}, {249, 14838}, {250, 32673}, {261, 572}, {270, 37304}, {274, 2224}, {283, 23707}, {293, 23996}, {304, 2172}, {305, 30918}, {306, 38822}, {310, 30895}, {319, 40589}, {320, 20769}, {326, 610}, {333, 909}, {409, 2646}, {448, 1944}, {513, 17940}, {560, 1740}, {584, 17379}, {593, 5035}, {604, 3759}, {644, 4606}, {648, 653}, {649, 17929}, {651, 1414}, {655, 4552}, {656, 1101}, {658, 1461}, {660, 765}, {661, 2644}, {668, 37218}, {670, 4586}, {689, 787}, {691, 8691}, {741, 3248}, {775, 820}, {798, 37134}, {799, 3570}, {805, 30670}, {811, 823}, {827, 831}, {849, 3216}, {892, 35180}, {894, 2174}, {908, 18653}, {911, 1444}, {922, 1580}, {932, 43359}, {985, 40734}, {997, 17512}, {1014, 41610}, {1018, 4596}, {1019, 3257}, {1021, 36141}, {1043, 2360}, {1086, 24617}, {1100, 1963}, {1125, 15792}, {1155, 2651}, {1169, 39979}, {1171, 30581}, {1193, 2363}, {1306, 6136}, {1307, 6135}, {1325, 4511}, {1333, 18274}, {1375, 37796}, {1412, 41629}, {1429, 24378}, {1434, 43762}, {1442, 40582}, {1474, 14013}, {1492, 1576}, {1509, 4251}, {1577, 15455}, {1625, 23113}, {1633, 4236}, {1691, 15994}, {1780, 37288}, {1781, 18714}, {1812, 1817}, {1818, 23692}, {1914, 2106}, {1924, 24037}, {1930, 34055}, {1933, 2227}, {1959, 2173}, {1966, 9417}, {1973, 18156}, {2126, 21081}, {2134, 21879}, {2167, 14213}, {2183, 17209}, {2194, 13588}, {2244, 17799}, {2245, 19308}, {2267, 17335}, {2268, 4687}, {2304, 31997}, {2326, 37448}, {2327, 8822}, {2341, 16578}, {2407, 38340}, {2421, 4603}, {2503, 24504}, {2576, 2580}, {2577, 2581}, {2635, 23695}, {2669, 9454}, {2748, 12074}, {2893, 28755}, {2939, 18719}, {2948, 4736}, {3109, 10609}, {3110, 3271}, {3204, 17350}, {3218, 37783}, {3219, 40592}, {3571, 5147}, {3573, 4436}, {3615, 35195}, {3682, 19842}, {3684, 17731}, {3737, 4570}, {3869, 37405}, {3939, 37138}, {4033, 7257}, {4069, 35368}, {4268, 17349}, {4276, 40110}, {4287, 17259}, {4473, 31059}, {4551, 36098}, {4559, 43069}, {4562, 4577}, {4563, 37215}, {4589, 37207}, {4590, 20981}, {4593, 4602}, {4594, 17941}, {4598, 34071}, {4599, 34072}, {4600, 4607}, {4601, 36860}, {4604, 34073}, {4620, 4625}, {4664, 9310}, {4670, 40744}, {4676, 11104}, {4858, 14616}, {4921, 16723}, {5009, 18792}, {5057, 5196}, {5086, 37158}, {5224, 25447}, {5235, 43757}, {5333, 25383}, {5468, 37210}, {5539, 23648}, {5698, 35915}, {5949, 7332}, {6061, 7411}, {6224, 6740}, {6507, 18750}, {6514, 27398}, {6518, 37774}, {6578, 15322}, {6626, 17256}, {7045, 23090}, {7054, 37659}, {7058, 14829}, {7192, 37143}, {7256, 43290}, {7258, 24004}, {7321, 18162}, {7452, 36797}, {8043, 21891}, {8052, 34076}, {8286, 26141}, {8301, 25048}, {8625, 34996}, {8652, 43356}, {8684, 43357}, {8690, 8694}, {8701, 34594}, {8708, 43076}, {9070, 36069}, {10330, 42720}, {13178, 21043}, {14206, 36102}, {14210, 36150}, {14534, 14554}, {14543, 17136}, {14953, 17139}, {14999, 17069}, {15507, 37019}, {16521, 40773}, {16545, 18049}, {16546, 18715}, {16598, 21381}, {16610, 37791}, {16704, 37222}, {16751, 36087}, {17104, 25440}, {17190, 27065}, {17206, 37214}, {17227, 25940}, {17289, 41534}, {17346, 34016}, {17438, 17868}, {17745, 46194}, {18020, 22382}, {18048, 18143}, {18051, 34082}, {18157, 34081}, {18206, 37131}, {18315, 39177}, {18594, 18713}, {18595, 18716}, {18596, 18717}, {18597, 18718}, {18598, 18720}, {18599, 18721}, {18740, 36145}, {19559, 34070}, {20337, 44387}, {20932, 34066}, {20948, 46254}, {20975, 24500}, {21008, 40432}, {21252, 30995}, {21254, 24714}, {22054, 28287}, {23363, 23831}, {23582, 32670}, {23999, 36139}, {24000, 36092}, {24018, 36131}, {24267, 25688}, {24557, 25887}, {24582, 24619}, {24636, 25469}, {24723, 35916}, {27665, 28283}, {27834, 34080}, {28210, 40522}, {28753, 37382}, {28841, 36066}, {30576, 37680}, {30962, 44081}, {30966, 37208}, {31998, 35148}, {32669, 37136}, {32851, 37793}, {33116, 40605}, {33297, 37213}, {34990, 35069}, {35049, 44769}, {36034, 36083}, {36036, 36132}

▸ Functional Relations32 relations

Notes

Let JaJbJc be the excentral triangle and H the Stammler hyperbola. Let T be the tangential triangle, with respect to JaJbJc, of H. Then T and ABC are perspective, and their perspector is X662. (Randy Hutson, December 26, 2015)

Let La be the A-extraversion of line X1X21, and define Lb, Lc cyclically. Let A' = Lb ∩ Lc, and define B' and C' cyclically. The lines AA', BB', CC' concur in X662. (Randy Hutson, January 29, 2018)

Let La be the A-extraversion of line X7X21, and define Lb, Lc cyclically. Let A' = Lb ∩ Lc, and define B' and C' cyclically. The lines AA', BB', CC' concur in X662. (Randy Hutson, January 29, 2018)

Let La be the A-extraversion of line X8X21, and define Lb, Lc cyclically. Let A' = Lb ∩ Lc, and define B' and C' cyclically. The lines AA', BB', CC' concur in X662. (Randy Hutson, January 29, 2018)

Let La be the A-extraversion of line X10X21, and define Lb, Lc cyclically. Let A' = Lb ∩ Lc, and define B' and C' cyclically. The lines AA', BB', CC' concur in X662. (Randy Hutson, January 29, 2018)

Let La be the A-extraversion of line X21X36, and define Lb, Lc cyclically. Let A' = Lb ∩ Lc, and define B' and C' cyclically. The lines AA', BB', CC' concur in X662. (Randy Hutson, January 29, 2018)