X₂₀

DE LONGCHAMPS POINT

X₂₀ = anticomplement of X₄

5, 6, 7 3, 4, 5 Kepler Kimberling

X₂₀ (DE LONGCHAMPS POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Anticomplementary Triangle △K_aK_bK_c — vertices K_a, K_b, K_c (Formed by lines through A, B, C parallel to opposite sides)

• Concurrence Lines: K_a–X₂₀, K_b–X₂₀, K_c–X₂₀ concurring at X₂₀

• Central Lines: {2, 3} (Euler line)

• Related Centers: X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₂ (CENTROID)

Trilinears: \(\cos A - \cos B \cos C : \cos B - \cos C \cos A : \cos C - \cos A \cos B\)

Barycentrics: \(\tan B + \tan C - \tan A : \tan C + \tan A - \tan B : \tan A + \tan B - \tan C\)

Tripolars: \(\sqrt{a^{6}-3 a^{2} b^{4}+2 b^{6}+6 a^{2} b^{2} c^{2}-2 b^{4} c^{2}-3 a^{2} c^{4}-2 b^{2} c^{4}+2 c^{6}} : :\)

Search Value (6, 9, 13): \(19.241345199319\)

▸ Alternate Coordinates5 forms

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

• Trilinears: \(2 \cos A - \sin B \sin C : 2 \cos B - \sin C \sin A : 2 \cos C - \sin A \sin B\)

• Trilinears: \((\csc A)(\tan B + \tan C - \tan A) : :\)

• Barycentrics: \([-3a^{4} + 2a^{2}(b^{2} + c^{2}) + (b^{2} - c^{2})^{2}] : :\)

• Barycentrics: \(S^{2} - 2 S_{B} S_{C} : :\)

▸ Linear Combinations2 combinations

• X₂₀ = \(2(r + 2R)\) X₁ - \((r +4R)\) X₇ = \(3\) X₂ - \(4\) X₃

• X₂₀ = \((1 - J)\) X₁₁₁₃ + \((1 + J)\) X₁₁₁₄

X₂₀ lies on the the following curves: Q046, Q063, Q070, Q073, Q115, K004, K007, K032, K041, K047, K071, K077, K080, K096, K099, K122, K169, K182, K236, K268, K270, K313, K329, K344, K364, K401, K425, K426, K443, K449, K462, K499, K522, K566, K609, K617, K648, K649, K650, K651, K652, K706, K753, K763, K778, K809, K814, K824, K825, K827, K850, K894. Euler-Gergonne-Soddy circle, GEOS circle, Steiner/Wallace rectangular hyperbola, anticomplement of Kiepert hyperbola, anticomplement of Feuerbach hyperbola, anticomplement of Jerabek hyperbola.

▸ Incident Central Lines480 lines

{1, 7}, {2, 3}, {6, 6459}, {8, 40}, {9, 10429}, {10, 165}, {11, 5204}, {12, 5217}, {13, 5238}, {14, 5237}, {15, 3412}, {16, 3411}, {17, 5352}, {18, 5351}, {32, 2549}, {33, 1038}, {34, 1040}, {35, 1478}, {36, 1479}, {39, 7737}, {46, 10572}, {51, 9729}, {52, 5890}, {54, 4846}, {55, 388}, {56, 497}, {57, 938}, {58, 387}, {61, 10653}, {62, 10654}, {64, 69}, {65, 3474}, {68, 74}, {72, 144}, {76, 3424}, {78, 329}, {81, 5706}, {97, 1217}, {98, 148}, {99, 147}, {100, 153}, {101, 152}, {103, 150}, {104, 149}, {107, 3184}, {109, 151}, {110, 146}, {112, 10316}, {113, 10721}, {114, 7912}, {115, 5206}, {116, 10725}, {117, 10726}, {118, 10727}, {119, 10728}, {120, 10729}, {121, 10730}, {122, 10152}, {123, 10731}, {124, 10732}, {125, 10733}, {126, 10734}, {127, 10735}, {142, 5436}, {145, 517}, {154, 2883}, {155, 323}, {159, 2139}, {172, 9598}, {182, 7787}, {184, 9545}, {185, 193}, {187, 3767}, {190, 1265}, {192, 9962}, {200, 5815}, {212, 1935}, {216, 3087}, {220, 5781}, {222, 3562}, {224, 4511}, {226, 3601}, {227, 9371}, {230, 5023}, {243, 1118}, {254, 1300}, {262, 5395}, {265, 11270}, {284, 5746}, {298, 5868}, {299, 5869}, {316, 7763}, {325, 6337}, {333, 5786}, {343, 6247}, {345, 7270}, {346, 1766}, {348, 4872}, {355, 3579}, {371, 1587}, {372, 1588}, {385, 6392}, {386, 9535}, {389, 3060}, {391, 573}, {392, 9856}, {393, 577}, {394, 1032}, {395, 5339}, {396, 5340}, {399, 6188}, {476, 2693}, {477, 10420}, {484, 10573}, {485, 1131}, {486, 1132}, {487, 638}, {488, 637}, {495, 9655}, {496, 9668}, {498, 3585}, {499, 3583}, {518, 3189}, {519, 5493}, {524, 11148}, {527, 11523}, {529, 3913}, {535, 5537}, {541, 9143}, {542, 8591}, {543, 7751}, {551, 11522}, {553, 11518}, {568, 10263}, {574, 2548}, {576, 5032}, {578, 5012}, {579, 5802}, {590, 6409}, {597, 10541}, {601, 3072}, {602, 3073}, {603, 1936}, {610, 8804}, {615, 6410}, {616, 633}, {617, 634}, {620, 7825}, {621, 627}, {622, 628}, {648, 9530}, {650, 8142}, {651, 7078}, {653, 3176}, {664, 7973}, {671, 11623}, {691, 2697}, {754, 7758}, {901, 2734}, {908, 4855}, {910, 6554}, {936, 1750}, {942, 3488}, {946, 3576}, {952, 3621}, {956, 5082}, {958, 2550}, {960, 5698}, {986, 11031}, {999, 1058}, {1001, 8273}, {1007, 7773}, {1056, 3295}, {1060, 6198}, {1062, 1870}, {1074, 1838}, {1075, 5667}, {1076, 1785}, {1078, 7616}, {1104, 4000}, {1124, 9660}, {1125, 1699}, {1141, 11671}, {1147, 1614}, {1151, 3068}, {1152, 3069}, {1154, 11271}, {1155, 1788}, {1160, 10784}, {1161, 10783}, {1176, 10548}, {1181, 1993}, {1204, 1899}, {1210, 3586}, {1212, 5819}, {1216, 4549}, {1249, 3172}, {1290, 2694}, {1293, 2370}, {1296, 2373}, {1320, 10305}, {1327, 3590}, {1328, 3591}, {1330, 3430}, {1335, 9647}, {1340, 2542}, {1341, 2543}, {1342, 2546}, {1343, 2547}, {1351, 7839}, {1352, 2896}, {1376, 2551}, {1384, 5305}, {1385, 3622}, {1394, 5930}, {1420, 9580}, {1440, 1804}, {1445, 5809}, {1453, 5222}, {1482, 3623}, {1483, 8148}, {1499, 6563}, {1511, 7728}, {1519, 4881}, {1568, 11202}, {1578, 3092}, {1579, 3093}, {1610, 1633}, {1619, 9914}, {1621, 11496}, {1632, 2892}, {1689, 2545}, {1690, 2544}, {1697, 10106}, {1698, 10164}, {1706, 5795}, {1729, 5011}, {1743, 10443}, {1764, 10449}, {1768, 9803}, {1834, 4252}, {1836, 2646}, {1853, 6696}, {1857, 1940}, {1891, 10319}, {1902, 7718}, {1914, 9597}, {1992, 8550}, {1994, 7592}, {2077, 5080}, {2128, 3685}, {2130, 2131}, {2287, 5776}, {2420, 6794}, {2456, 10131}, {2482, 7888}, {2781, 6293}, {2782, 5984}, {2797, 9409}, {2800, 6224}, {2801, 5904}, {2822, 2939}, {2823, 4552}, {2888, 3357}, {2893, 10432}, {2894, 2975}, {2899, 5205}, {2917, 2935}, {2944, 3923}, {2947, 3682}, {2979, 5562}, {3047, 5504}, {3053, 5254}, {3054, 5585}, {3057, 3476}, {3058, 3304}, {3062, 5785}, {3095, 7709}, {3180, 5865}, {3181, 5864}, {3182, 3347}, {3183, 3348}, {3218, 5709}, {3219, 3587}, {3241, 5882}, {3244, 11531}, {3278, 3608}, {3303, 5434}, {3311, 7581}, {3312, 7582}, {3313, 5596}, {3316, 6451}, {3317, 6452}, {3333, 10580}, {3334, 3609}, {3339, 6738}, {3353, 3354}, {3355, 3637}, {3359, 5554}, {3361, 11019}, {3398, 10788}, {3419, 3916}, {3421, 5687}, {3431, 3521}, {3452, 5438}, {3472, 3473}, {3475, 10404}, {3564, 7893}, {3567, 5446}, {3598, 3673}, {3618, 5085}, {3619, 10516}, {3624, 3817}, {3634, 7989}, {3635, 11224}, {3648, 5693}, {3655, 10222}, {3666, 5716}, {3667, 5592}, {3697, 9947}, {3734, 7800}, {3788, 7842}, {3796, 11425}, {3812, 10178}, {3813, 11194}, {3849, 7759}, {3869, 6001}, {3870, 6769}, {3871, 10306}, {3872, 9874}, {3876, 5777}, {3911, 5704}, {3917, 5907}, {3933, 10513}, {3935, 5534}, {3972, 7803}, {4257, 5292}, {4385, 7172}, {4640, 5794}, {4652, 5175}, {4678, 5690}, {4848, 5128}, {4857, 10072}, {5007, 7739}, {5013, 7736}, {5044, 5927}, {5126, 11373}, {5174, 6350}, {5208, 10441}, {5223, 6743}, {5226, 9612}, {5247, 9441}, {5250, 9800}, {5270, 10056}, {5303, 11680}, {5316, 9842}, {5318, 11480}, {5321, 11481}, {5328, 6700}, {5418, 6564}, {5420, 6565}, {5422, 10982}, {5432, 10588}, {5433, 10589}, {5439, 5806}, {5440, 5658}, {5441, 5902}, {5447, 5891}, {5450, 10527}, {5462, 9781}, {5550, 8227}, {5587, 6684}, {5601, 9834}, {5602, 9835}, {5640, 10110}, {5654, 7712}, {5663, 6101}, {5714, 11374}, {5720, 5811}, {5730, 10609}, {5749, 10445}, {5758, 5905}, {5766, 8545}, {5841, 10528}, {5853, 6762}, {5876, 10627}, {5893, 10192}, {6102, 6243}, {6146, 6515}, {6193, 6241}, {6197, 9536}, {6214, 10518}, {6215, 10517}, {6221, 7583}, {6249, 9751}, {6264, 9802}, {6326, 9809}, {6390, 7776}, {6398, 7584}, {6449, 8981}, {6455, 8976}, {6462, 6465}, {6463, 6466}, {6526, 11589}, {6680, 7872}, {6737, 7992}, {6744, 10980}, {6765, 7994}, {6766, 9797}, {6767, 10386}, {7074, 9370}, {7596, 10885}, {7618, 7775}, {7620, 8182}, {7694, 7752}, {7730, 11802}, {7731, 11562}, {7749, 8588}, {7755, 11648}, {7761, 7795}, {7768, 11057}, {7774, 7783}, {7784, 7789}, {7785, 9737}, {7797, 9753}, {7799, 7860}, {7801, 7873}, {7818, 7863}, {7820, 7935}, {7832, 7910}, {7835, 7911}, {7836, 7898}, {7864, 9748}, {7885, 7891}, {7921, 10983}, {7971, 11682}, {7998, 11439}, {8069, 10629}, {8081, 9793}, {8082, 9795}, {8111, 9783}, {8112, 9787}, {8117, 8118}, {8119, 8124}, {8120, 8123}, {8164, 9654}, {8234, 9789}, {8235, 9791}, {8726, 9776}, {8861, 9474}, {8983, 9615}, {9529, 9979}, {9786, 11433}, {9927, 11468}, {9957, 11035}, {9993, 10583}, {9996, 10357}, {10246, 10595}, {10267, 10532}, {10269, 10531}, {10282, 11449}, {10359, 10796}, {10453, 10476}, {10470, 10478}, {10525, 10785}, {10526, 10786}, {10543, 11246}, {10584, 10893}, {10585, 10894}, {10601, 11745}, {10679, 10805}, {10680, 10806}, {11180, 11645}, {11451, 11695}, {11470, 11511}, {11472, 11487}, {11473, 11513}, {11474, 11514}, {11475, 11515}, {11476, 11516}

▸ Functional Relations80 relations

• X₂₀ = 9 X₂ - 8 X₅ = 3 X₄ - 4 X₅ = 3 X₃ - 2 X₅ = 15 X₂ - 16 X₁₄₀ = 5 X₄ - 8 X₁₄₀ = 5 X₅ - 6 X₁₄₀ = 5 X₃ - 4 X₁₄₀ = 2 X₁₀ - 3 X₁₆₅ = 8 X₁₄₀ - 15 X₃₇₆ = 4 X₅ - 9 X₃₇₆ = 2 X₃ - 3 X₃₇₆ = X₄ - 3 X₃₇₆ = 10 X₅ - 9 X₃₈₁ = 5 X₄ - 6 X₃₈₁ = 5 X₂ - 4 X₃₈₁ = 5 X₃ - 3 X₃₈₁ = 4 X₁₄₀ - 3 X₃₈₁ = 5 X₃₇₆ - 2 X₃₈₁ = 12 X₁₄₀ - 5 X₃₈₂ = 9 X₃₈₁ - 5 X₃₈₂ = 9 X₂ - 4 X₃₈₂ = 3 X₄ - 2 X₃₈₂ = 9 X₃₇₆ - 2 X₃₈₂ = 3 X₃ - X₃₈₂ = 7 X₃₈₂ - 12 X₅₄₆

• X₂₀ = midpoint of X_i and X_j for these {i,j}: {3, 1657}, {4, 3529}, {376, 11001}, {944, 6361}, {1498, 5925}, {3146, 5059}, {3869, 9961}, {6241, 11412}, {10575, 10625}

• X₂₀ = reflection of X_i in X_j for these (i,j): (1, 4297), (2, 376), (3, 550), (4, 3), (5, 548), (7, 5732), (8, 40), (23, 10295), (64, 5894), (65, 9943), (68, 7689), (69, 1350), (76, 5188), (107, 3184), (144, 5759), (145, 944), (146, 110), (147, 99), (148, 98), (149, 104), (150, 103), (151, 109), (152, 101), (153, 100), (176, 8984), (193, 6776), (194, 11257), (329, 6282), (355, 3579), (376, 3534), (381, 8703), (382, 5), (616, 5473), (617, 5474), (650, 8142), (938, 9841), (962, 1), (1330, 3430), (1352, 3098), (1375, 8153), (2475, 3651), (2550, 11495), (2888, 7691), (3091, 3522), (3146, 4), (3153, 2071), (3421, 6244), (3434, 3428), (3436, 10310), (3448, 74), (3529, 1657), (3543, 2), (3627, 140), (3830, 549), (3832, 3528), (3839, 10304), (3853, 3530), (3868, 1071), (4846, 8717), (5059, 3529), (5073, 3627), (5080, 2077), (5189, 7464), (5691, 10), (5768, 7171), (5876, 10627), (5878, 6759), (5881, 11362), (5889, 185), (5895, 2883), (5921, 69), (5984, 9862), (6223, 1490), (6225, 1498), (6241, 10575), (6243, 6102), (6256, 6796), (6515, 10605), (6655, 7470), (6764, 6762), (6839, 7411), (6840, 6909), (6895, 6906), (6925, 7580), (7379, 4229), (7391, 378), (7408, 3537), (7620, 8182), (7710, 8719), (7728, 1511), (7731, 11562), (7758, 7781), (7982, 5882), (7991, 5493), (8148, 1483), (9589, 4301), (9797, 9845), (9799, 84), (9802, 6264), (9803, 1768), (9809, 6326), (9812, 3576), (9863, 7750), (9965, 2096), (10152, 122), (10296, 858), (10431, 1012), (10446, 991), (10721, 113), (10722, 114), (10723, 115), (10724, 11), (10725, 116), (10726, 117), (10727, 118), (10728, 119), (10729, 120), (10730, 121), (10731, 123), (10732, 124), (10733, 125), (10734, 126), (10735, 127), (10736, 1313), (10737, 1312), (11185, 8722), (11381, 5907), (11412, 10625), (11455, 5891), (11477, 8550), (11531, 3244), (11541, 5073), (11671, 1141)

• X₂₀ = isogonal conjugate of X₆₄

• X₂₀ = isotomic conjugate of X₂₅₃

• X₂₀ = cyclocevian conjugate of X₁₀₃₂

• X₂₀ = complement of X₃₁₄₆

• X₂₀ = anticomplement of X₄

• X₂₀ = anticomplementary conjugate of X₄

• X₂₀ = X_i-Ceva conjugate of X_j for these (i,j): (69, 2), (489, 487), (490, 488), (801, 6), (1043, 1), (1350, 6194), (1503, 147), (1975, 194), (5921, 9742), (7750, 2896), (8822, 63)

• X₂₀ = X_i-cross conjugate of X_j for these (i,j): (64, 2131), (122, 8057), (154, 1249), (1249, 2), (1498, 6616), (3183, 2060), (3198, 610), (5895, 4), (5930, 1895), (6525, 3344)

• X₂₀ = crosspoint of X₁ and X₇₀₃₈

• X₂₀ = crosssum of X_i and X_j for these (i,j): {1, 1044}, {512, 3269}, {649, 3270}

• X₂₀ = crossdifference of every pair of points on line X₆₄₇X₆₅₇

• X₂₀ = trilinear pole of X₆₅₈₇X_8057

• X₂₀ = circumcircle-inverse of X₂₀₇₁

• X₂₀ = orthocentroidal-circle-inverse of X₃₀₉₁

• X₂₀ = Steiner-circle-inverse of X₈₅₈

• X₂₀ = polar-circle-inverse of X₁₀₁₅₁

• X₂₀ = orthoptic-circle-of-Steiner-inellipse-inverse of X₅₁₅₉

• X₂₀ = orthoptic-circle-of-Steiner-circumelipse-inverse of X₈₅₈

• X₂₀ = Grebe-circle-inverse of X₃₇₉₄₄

• X₂₀ = Johnson-circle-inverse of anticomplement of X₃₇₉₆₈

• X₂₀ = anticomplement-of-circumcircle-inverse of X₃₁₅₃

• X₂₀ = X_i-aleph conjugate of X_j for these (i,j): (8, 191), (9, 1045), (21, 3216), (29, 1714), (188, 1046), (333, 2), (556, 1762), (645, 3882), (1043, 20), (3699, 4427), (4182, 846), (6731, 2938)

• X₂₀ = X_i-beth conjugate of X_j for these (i,j): (8, 5691), (20, 1394), (21, 4306), (643, 1259), (664, 20), (1043, 280)

• X₂₀ = X_i-gimel conjugate of X_j for these (i,j): (21, 6848), (1792, 20), (3900, 20), (4397, 20), (7253, 20)

• X₂₀ = X_i-he conjugate of X_j for these (i,j): (645, 20), (799, 20), (7256, 20), (7258, 20)

• X₂₀ = X_i-zayin conjugate of X_j for these (i,j): (1, 64), (200, 7580), (1043, 20), (2287, 573), (4397, 3667), (6737, 40)

• X₂₀ = antigonal conjugate of X₁₀₁₅₂

• X₂₀ = syngonal conjugate of X₃₁₈₄

• X₂₀ = X_i-anticomplementary conjugate of X_j for these (i,j): (1, 4), (3, 8), (4, 5906), (6, 5905), (19, 6515), (31, 193), (47, 6193), (48, 2), (55, 5942), (58, 3868), (63, 69), (69, 6327), (71, 2895), (72, 1330), (73, 2475), (75, 11442), (77, 3434), (78, 3436), (82, 3060), (91, 68), (92, 317), (101, 4391), (102, 5081), (109, 521), (110, 7253), (162, 520), (163, 525), (184, 192), (212, 144), (219, 329), (222, 7), (228, 1654), (255, 20), (268, 189), (283, 3869), (284, 92), (293, 511), (295, 4645), (304, 315), (326, 1370), (394, 4329), (577, 6360), (603, 145), (656, 3448), (662, 850), (810, 148), (905, 150), (906, 514), (921, 11411), (922, 7665), (947, 318), (951, 5174), (1069, 11415), (1101, 110), (1110, 3732), (1214, 2893), (1262, 4566), (1331, 513), (1333, 3187), (1433, 962), (1437, 1), (1459, 149), (1496, 11469), (1790, 75), (1794, 72), (1795, 517), (1796, 319), (1797, 320), (1803, 85), (1807, 5080), (1813, 693), (1815, 4872), (1822, 2574), (1823, 2575), (1923, 10340), (1964, 8878), (1973, 6392), (2148, 1993), (2149, 651), (2159, 3580), (2164, 2994), (2167, 264), (2168, 5392), (2169, 3), (2190, 5889), (2193, 63), (2196, 6542), (2200, 1655), (2216, 52), (2349, 340), (2359, 321), (2360, 1895), (2576, 2592), (2577, 2593), (2964, 11271), (3916, 2891), (3990, 3151), (4020, 2896), (4303, 2894), (4558, 7192), (4575, 523), (4587, 4462), (4592, 512), (6507, 6527), (7011, 5932), (7015, 4388), (7078, 6223), (7099, 4452), (7116, 6646), (7125, 347), (7177, 6604), (9247, 194), (9255, 1899)

• X₂₀ = X₃₅₃₂-complementary conjugate of X₁₀

• X₂₀ = X_i-vertex conjugate of X_j for these (i,j): {3, 3346}, {4, 5879}, {523, 2071}

• X₂₀ = X₄-of-anticomplementary triangle

• X₂₀ = X₅₂-of-hexyl-triangle

• X₂₀ = reflection of X₁₀₂₉₆ in the De Longchamps line

• X₂₀ = perspector of anticomplementary triangle and polar triangle of de Longchamps circle

• X₂₀ = isogonal conjugate of X₄ wrt anticevian triangle of X₄ (or ‘anticevian-isogonal conjugate of X₄’)

• X₂₀ = perspector of ABC and pedal triangle of X₁₄₉₈

• X₂₀ = exsimilicenter of circumcircle and 1st Steiner circle (the insimilicenter is X₆₃₁)

• X₂₀ = X₄-of-circumcevian-triangle-of-X₃₀

• X₂₀ = anticomplementary isotomic conjugate of X₁₉₃

• X₂₀ = excentral isogonal conjugate of X₁₀₄₆

• X₂₀ = excentral isotomic conjugate of X₁₀₄₅

• X₂₀ = cevapoint of X_i and X_j for these {i,j}: {1, 3182}, {3, 1498}, {4, 3183}, {6, 1661}, {30, 3184}, {40, 1490}, {64, 2130}, {84, 3353}, {122, 8057}, {577, 1660}, {610, 7070}, {1249, 3079}, {3198, 8804}, {3345, 3472}, {3346, 3355}

• X₂₀ = radical center of power circles

• X₂₀ = radical center of circles centered at the vertices of ABC with radius equal to opposite side

• X₂₀ = intersection of tangents to conic {X₄,X₁₃,X₁₄,X₁₅,X₁₆} at X₁₅ and X₁₆

• X₂₀ = trilinear pole wrt anticomplementary triangle of de Longchamps line

• X₂₀ = trilinear pole of polar of X₄₅₉ wrt polar circle, which is also the perspectrix of ABC and the half-altitude triangle

• X₂₀ = pole wrt polar circle of trilinear polar of X₄₅₉

• X₂₀ = isoconjugate of X_j and X_j for these (i,j): {1, 64}, {2, 2155}, {6, 2184}, {19, 1073}, {31, 253}, {48, 459}, {55, 8809}, {255, 6526}, {656, 1301}, {1036, 10375}, {1402, 5931}, {2190, 8798}

• X₂₀ = circumcevian isogonal conjugate of X₃

• X₂₀ = perspector of ABC and circumcircle antipode of circumanticevian triangle of X₄

• X₂₀ = X₉₈-of-6th-Brocard-triangle

• X₂₀ = perspector of hexyl triangle and cross-triangle of ABC and hexyl triangle

• X₂₀ = Thomson isogonal conjugate of X₃₁₆₇

• X₂₀ = Lucas isogonal conjugate of X₂

• X₂₀ = inner-Conway-to-Conway similarity image of X₈

• X₂₀ = cyclocevian conjugate of X₂ wrt anticevian triangle of X₂

• X₂₀ = trilinear product of vertices of X₄-anti-altimedial triangle

• X₂₀ = homothetic center of X₂₀-altimedial and X₂-anti-altimedial triangles

• X₂₀ = inverse-in-orthoptic-circle-of-Steiner-inellipse of X₅₁₅₉

• X₂₀ = inverse-in-circumconic-centered-at-X₄ of X₁₅₅₉

• X₂₀ = anticevian isogonal conjugate of X₄

• X₂₀ = X₅₅₆₂-of-excentral-triangle

• X₂₀ = X₇₄-of-X₃-Fuhrmann-triangle

• X₂₀ = Ehrmann-mid-to-Johnson similarity image of X₃

• X₂₀ = perspector of hexyl triangle and anticevian triangle of X₆₃

• X₂₀ = perspector of excentral triangle and tangential triangle wrt hexyl triangle of the excentral-hexyl ellipse

• X₂₀ = perspector of excentral triangle and extraversion triangle of X₇

• X₂₀ = homothetic center of ABC and the reflection in X₃ of the pedal triangle of X₃ (medial triangle)

• X₂₀ = homothetic center of ABC and the reflection in X₄ of the antipedal triangle of X₄ (anticomplementary triangle)

• X₂₀ = orthic-isogonal conjugate of X₃₂₆₀₅

• X₂₀ = pole of Brocard axis wrt conic {X₄,X₁₃,X₁₄,X₁₅,X₁₆}}

• X₂₀ = QA-P₅ (Isotomic Center) of the incenter-excenters quadrangle (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/26-qa-p5.html)

• X₂₀ = barycentric product X_i*X_j for these {i,j}: {63, 1895}, {69, 1249}, {75, 610}, {76, 154}, {85, 7070}, {86, 8804}, {99, 6587}, {204, 304}, {274, 3198}, {305, 3172}, {312, 1394}, {333, 5930}, {648, 8057}, {801, 2883}, {1032, 6616}, {1097, 2184}, {3213, 3718}, {3344, 6527}, {3926, 6525}, {7156, 7182}, {10152, 11064}

• X₂₀ = barycentric quotient X_i/X_j for these (i,j): (1, 2184), (2, 253), (3, 1073), (4, 459), (6, 64), (31, 2155), (57, 8809), (112, 1301), (154, 6), (204, 19), (216, 8798), (333, 5931), (393, 6526), (610, 1), (1249, 4), (1394, 57), (1498, 3343), (1562, 125), (1895, 92), (2285, 10375), (3079, 1249), (3172, 25), (3198, 37), (3213, 34), (3284, 11589), (3344, 3346), (5930, 226), (6525, 393), (6587, 523), (7070, 9), (7156, 33), (8057, 525), (8804, 10)

• X₂₀ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): (1, 7, 11036), (1, 1044, 1042), (1, 1742, 4300), (1, 1770, 4295), (1, 3100, 9538), (1, 4292, 7), (1, 4293, 3600), (1, 4294, 390), (1, 4297, 5731), (1, 4298, 11037), (1, 4299, 4293), (1, 4301, 5734), (1, 4302, 4294), (1, 4304, 4313), (1, 4311, 4308), (1, 4312, 3671), (1, 4316, 4299), (1, 4324, 4302), (1, 4325, 4317), (1, 4330, 4309), (1, 4333, 1770), (1, 4340, 3945), (1, 4355, 5542), (1, 5732, 10884), (1, 9589, 4301), (1, 10624, 9785), (2, 3, 3523), (2, 4, 3091), (2, 5, 7486), (2, 22, 10565), (2, 23, 4232), (2, 376, 10304), (2, 377, 4208), (2, 452, 5129), (2, 1370, 7396), (2, 2475, 5177), (2, 3091, 5056), (2, 3146, 4), (2, 3522, 3), (2, 3523, 10303), (2, 3543, 3839), (2, 3832, 5), (2, 4190, 6904), (2, 5046, 6919), (2, 5059, 3146), (2, 5068, 3090), (2, 6837, 6884), (2, 6838, 6960), (2, 6839, 6993), (2, 6847, 6888), (2, 6848, 6979), (2, 6872, 452), (2, 6890, 6972), (2, 6987, 6992), (2, 6995, 7398), (2, 7391, 7378), (2, 7408, 6997), (2, 7409, 5133), (2, 7500, 6995), (2, 11106, 405), (3, 4, 2), (3, 5, 631), (3, 22, 7488), (3, 26, 186), (3, 140, 3524), (3, 376, 3522), (3, 381, 140), (3, 382, 5), (3, 405, 6986), (3, 546, 3525), (3, 548, 3528), (3, 549, 10299), (3, 550, 376), (3, 1012, 21), (3, 1532, 6921), (3, 1593, 7503), (3, 1597, 7395), (3, 1656, 549), (3, 1885, 6816), (3, 2937, 1658), (3, 3079, 2060), (3, 3091, 10303), (3, 3146, 3091), (3, 3149, 404), (3, 3522, 10304), (3, 3526, 3530), (3, 3529, 3146), (3, 3534, 550), (3, 3543, 5056), (3, 3560, 1006), (3, 3575, 6815), (3, 3627, 3090), (3, 3830, 1656), (3, 3843, 3526), (3, 3845, 3533), (3, 3851, 5054), (3, 3853, 5067), (3, 5059, 3543), (3, 5073, 381), (3, 5076, 3628), (3, 6638, 417), (3, 6756, 6803), (3, 6823, 7494), (3, 6827, 6926), (3, 6831, 6910), (3, 6836, 6890), (3, 6840, 6972), (3, 6842, 6954), (3, 6850, 6908), (3, 6851, 6847), (3, 6868, 6987), (3, 6872, 6992), (3, 6882, 6961), (3, 6895, 6888), (3, 6905, 4188), (3, 6906, 4189), (3, 6907, 6988), (3, 6911, 6940), (3, 6914, 6875), (3, 6917, 6889), (3, 6923, 6825), (3, 6925, 6838), (3, 6928, 6891), (3, 6929, 6967), (3, 6934, 4190), (3, 6938, 6872), (3, 6985, 6905), (3, 7387, 24), (3, 7395, 7485), (3, 7488, 10298), (3, 7491, 6827), (3, 7517, 6644), (3, 7553, 7401), (3, 7580, 411), (3, 8727, 6857), (3, 9122, 1817), (3, 9818, 7509), (3, 9909, 3515), (3, 10323, 6636), (3, 10431, 6837), (3, 11001, 5059), (3, 11413, 2071), (3, 11414, 22), (3, 11479, 7484), (3, 11676, 3552), (4, 5, 3832), (4, 21, 6837), (4, 24, 3089), (4, 140, 5068), (4, 186, 3542), (4, 376, 3), (4, 377, 6839), (4, 378, 3088), (4, 404, 6953), (4, 411, 6838), (4, 443, 6835), (4, 550, 3522), (4, 631, 5), (4, 1006, 6846), (4, 1595, 7409), (4, 1656, 3854), (4, 1657, 5059), (4, 3079, 6616), (4, 3088, 7378), (4, 3090, 381), (4, 3091, 3839), (4, 3146, 3543), (4, 3147, 403), (4, 3520, 3541), (4, 3522, 3523), (4, 3523, 5056), (4, 3524, 3090), (4, 3525, 3545), (4, 3528, 631), (4, 3533, 3851), (4, 3537, 6803), (4, 3538, 6804), (4, 3542, 6623), (4, 3545, 546), (4, 3651, 6908), (4, 3855, 3843), (4, 4188, 6979), (4, 4189, 6888), (4, 5067, 3855), (4, 5084, 6957), (4, 6353, 235), (4, 6616, 1559), (4, 6756, 7408), (4, 6803, 6997), (4, 6815, 7544), (4, 6824, 6870), (4, 6825, 6871), (4, 6826, 6894), (4, 6827, 5046), (4, 6833, 6844), (4, 6836, 6840), (4, 6850, 2475), (4, 6851, 6895), (4, 6852, 6866), (4, 6853, 6867), (4, 6854, 6849), (4, 6856, 7548), (4, 6857, 6828), (4, 6865, 2478), (4, 6868, 6872), (4, 6875, 6824), (4, 6876, 6825), (4, 6878, 6990), (4, 6880, 6941), (4, 6889, 6843), (4, 6891, 5187), (4, 6897, 6826), (4, 6899, 6827), (4, 6902, 6929), (4, 6903, 6928), (4, 6905, 6848), (4, 6906, 6847), (4, 6908, 5177), (4, 6909, 6890), (4, 6916, 377), (4, 6926, 6919), (4, 6927, 1532), (4, 6935, 6831), (4, 6940, 6964), (4, 6942, 6834), (4, 6947, 6893), (4, 6948, 4190), (4, 6949, 6968), (4, 6950, 6833), (4, 6951, 6917), (4, 6954, 5141), (4, 6961, 5154), (4, 6977, 6830), (4, 6986, 6886), (4, 6987, 452), (4, 6988, 2476), (4, 7386, 6816), (4, 7401, 7394), (4, 7412, 4194), (4, 7487, 6995), (4, 7512, 3547), (4, 7714, 5198), (4, 8889, 7507), (4, 10299, 1656), (4, 10304, 10303), (4, 10323, 7400), (4, 10996, 6815), (4, 11001, 3529), (4, 11111, 6912), (5, 140, 5070), (5, 382, 4), (5, 548, 3), (5, 550, 548), (5, 631, 2), (5, 3526, 5067), (5, 3530, 3526), (5, 3627, 3861), (5, 3832, 3091), (5, 3843, 3855), (5, 3845, 3859), (5, 3853, 3843), (5, 3859, 3851), (5, 3861, 381), (5, 5070, 3090), (5, 7486, 5056), (5, 9715, 7493), (7, 3188, 279), (7, 4313, 1), (8, 9778, 40), (8, 10430, 9799), (8, 10538, 280), (11, 5204, 7288), (12, 5217, 5218), (21, 377, 2), (21, 6839, 6884), (21, 7411, 3), (22, 858, 7493), (22, 1370, 2), (22, 2071, 10298), (22, 11413, 3), (24, 3089, 4232), (24, 6643, 2), (24, 7387, 23), (25, 1885, 4), (25, 7386, 2), (25, 7667, 7386), (27, 464, 2), (32, 2549, 5286), (32, 5286, 5304), (32, 7756, 2549), (32, 7765, 5319), (35, 1478, 3085), (35, 3085, 5281), (35, 10483, 1478), (36, 1479, 3086), (36, 3086, 5265), (40, 84, 63), (40, 3101, 9537), (40, 5881, 11362), (55, 7354, 388), (56, 6284, 497), (57, 950, 938), (65, 5918, 9943), (68, 11457, 3448), (69, 6527, 253), (76, 5188, 6194), (98, 5171, 7793), (99, 315, 3926), (99, 7802, 315), (100, 3436, 7080), (140, 381, 3090), (140, 546, 10109), (140, 3090, 2), (140, 3627, 381), (140, 3861, 5), (140, 5073, 4), (140, 8703, 3), (145, 9965, 3868), (154, 5895, 2883), (165, 5691, 10), (175, 176, 347), (187, 7748, 3767), (226, 3601, 5703), (235, 3515, 6353), (315, 8721, 147), (316, 7782, 7763), (355, 3579, 5657), (355, 5657, 3617), (371, 1587, 7585), (371, 6560, 1587), (372, 1588, 7586), (372, 6561, 1588), (376, 631, 3528), (376, 1370, 2071), (376, 1657, 3146), (376, 3146, 3523), (376, 3524, 8703), (376, 3528, 548), (376, 3529, 4), (376, 5059, 3091), (376, 6240, 7400), (376, 6851, 4189), (376, 6869, 4190), (376, 6916, 7411), (376, 6938, 6987), (376, 11541, 140), (377, 1012, 6837), (377, 6837, 6993), (377, 10431, 4), (378, 6240, 4), (378, 10323, 3), (381, 3090, 5068), (381, 3524, 2), (381, 3627, 4), (381, 5068, 3091), (381, 5070, 5), (381, 5073, 3627), (381, 8703, 3524), (381, 10109, 3545), (382, 548, 631), (382, 550, 3528), (382, 631, 3832), (382, 3526, 3843), (382, 3528, 2), (382, 3530, 3855), (382, 3843, 3853), (382, 5070, 3861), (384, 7791, 2), (384, 7833, 7791), (390, 3600, 1), (394, 1498, 11441), (404, 2478, 2), (404, 11114, 2478), (405, 443, 2), (405, 6835, 6886), (405, 11111, 11106), (405, 11112, 443), (411, 6836, 2), (411, 6840, 6960), (411, 6909, 3), (411, 6943, 6962), (427, 7494, 2), (428, 7484, 7392), (428, 11403, 4), (440, 7490, 2), (442, 6857, 2), (442, 8727, 6828), (443, 6912, 6886), (443, 11111, 405), (452, 6904, 2), (474, 5084, 2), (474, 11113, 5084), (485, 6200, 9540), (485, 9540, 8972), (487, 638, 1271), (488, 637, 1270), (489, 490, 69), (498, 3585, 10590), (499, 3583, 10591), (546, 549, 1656), (546, 1656, 3545), (546, 3545, 3854), (546, 3830, 4), (546, 10299, 2), (547, 3858, 5072), (548, 3528, 3522), (548, 3853, 3530), (549, 1656, 3525), (549, 3545, 2), (549, 3830, 3545), (550, 1657, 4), (550, 3146, 10304), (550, 3529, 2), (550, 3627, 8703), (550, 5059, 3523), (550, 6240, 6636), (550, 11001, 3146), (574, 7747, 2548), (578, 10984, 5012), (631, 3528, 3), (631, 3832, 7486), (631, 3855, 5067), (631, 5067, 3526), (632, 3850, 5055), (858, 7493, 2), (858, 10296, 3153), (944, 2096, 1071), (946, 3576, 3616), (958, 11495, 5584), (962, 5731, 1), (962, 5734, 4301), (962, 10884, 11036), (991, 3332, 3945), (1006, 6826, 2), (1011, 6817, 2), (1012, 6916, 2), (1042, 3000, 1044), (1092, 6759, 110), (1113, 1114, 2071), (1131, 8972, 485), (1147, 1614, 9544), (1151, 3070, 3068), (1152, 3071, 3069), (1155, 1837, 1788), (1352, 3098, 10519), (1352, 10519, 3620), (1368, 6353, 2), (1368, 9909, 6353), (1370, 7493, 858), (1385, 5603, 3622), (1478, 3085, 5261), (1479, 3086, 5274), (1482, 7967, 3623), (1490, 6282, 78), (1532, 6922, 4193), (1583, 6805, 2), (1584, 6806, 2), (1585, 1589, 2), (1586, 1590, 2), (1587, 9541, 371), (1593, 3575, 4), (1593, 7503, 7527), (1593, 10996, 2), (1595, 7399, 5133), (1597, 3537, 2), (1597, 6756, 4), (1598, 3538, 2), (1610, 1633, 3556), (1656, 3525, 2), (1656, 3830, 546), (1656, 10109, 3090), (1657, 3534, 3), (1657, 8703, 11541), (1658, 2937, 7556), (1699, 7987, 1125), (1764, 10454, 10449), (1764, 10463, 10461), (1836, 2646, 3485), (1853, 8567, 6696), (1975, 7750, 69), (2041, 2042, 4), (2043, 2044, 376), (2045, 2046, 3533), (2060, 3146, 1559), (2071, 7488, 3), (2071, 10296, 858), (2077, 6256, 5552), (2475, 4189, 2), (2475, 6847, 3091), (2475, 6895, 4), (2475, 6906, 6888), (2476, 6910, 2), (2478, 3149, 6953), (2479, 2480, 441), (2549, 5319, 7765), (3053, 5254, 7735), (3060, 10574, 389), (3088, 6636, 3523), (3088, 7400, 2), (3090, 3524, 140), (3090, 3529, 11541), (3090, 11541, 3627), (3091, 3523, 2), (3091, 3543, 4), (3091, 4208, 6993), (3091, 6992, 5129), (3091, 7486, 5), (3091, 10304, 3523), (3098, 9873, 2896), (3100, 4296, 1), (3146, 3522, 2), (3146, 3523, 3839), (3146, 3528, 7486), (3146, 3854, 3830), (3146, 7411, 4208), (3146, 10304, 5056), (3146, 11413, 7396), (3147, 11585, 2), (3149, 6865, 2), (3151, 7560, 2), (3152, 7538, 2), (3153, 10298, 2), (3474, 3486, 65), (3520, 7512, 3), (3522, 3529, 3543), (3522, 3543, 10303), (3522, 3854, 10299), (3522, 5059, 4), (3523, 3543, 3091), (3523, 10304, 3), (3524, 3529, 5073), (3524, 3627, 5068), (3524, 11541, 4), (3525, 3545, 1656), (3525, 3830, 3854), (3525, 10299, 549), (3526, 3530, 631), (3526, 3843, 5), (3526, 3853, 3855), (3526, 5067, 2), (3528, 3529, 382), (3528, 3832, 3523), (3528, 3855, 3530), (3529, 3534, 3522), (3529, 11001, 1657), (3530, 3843, 5067), (3530, 3853, 5), (3530, 3855, 2), (3533, 3628, 2), (3533, 5071, 3628), (3534, 11001, 2), (3541, 3547, 2), (3542, 3546, 2), (3543, 7396, 3153), (3543, 10304, 2), (3545, 3854, 3091), (3545, 10299, 3525), (3548, 7505, 2), (3552, 6655, 2), (3560, 6826, 6846), (3560, 6897, 2), (3567, 5446, 11002), (3575, 7503, 7544), (3583, 7280, 499), (3585, 5010, 498), (3601, 9579, 226), (3616, 9812, 946), (3627, 8703, 140), (3627, 11541, 3146), (3628, 3845, 3851), (3628, 3851, 5071), (3628, 3859, 5), (3628, 5054, 3533), (3651, 6851, 2), (3651, 6906, 3), (3734, 7830, 7800), (3839, 5056, 3091), (3839, 10303, 5056), (3843, 3853, 4), (3843, 3855, 3832), (3845, 5054, 5071), (3845, 5076, 4), (3851, 5054, 3628), (3851, 5076, 3845), (3853, 5067, 3832), (3855, 5067, 5), (3868, 11220, 1071), (3911, 9581, 5704), (3917, 5907, 11444), (3917, 11381, 5907), (3972, 7847, 7803), (4188, 5046, 2), (4188, 6926, 3523), (4189, 6895, 6847), (4189, 6908, 3523), (4190, 6868, 6992), (4190, 6872, 2), (4190, 6987, 3523), (4191, 6818, 2), (4193, 6921, 2), (4195, 4201, 2), (4197, 10883, 5), (4208, 6837, 5056), (4292, 4297, 10884), (4292, 4304, 1), (4292, 4313, 11036), (4293, 4294, 1), (4293, 4302, 390), (4294, 4299, 3600), (4295, 4305, 1), (4298, 4314, 1), (4299, 4302, 1), (4299, 4309, 4317), (4299, 4317, 4325), (4299, 4324, 4294), (4301, 9589, 962), (4302, 4309, 4330), (4302, 4316, 4293), (4302, 4317, 4309), (4304, 5732, 5731), (4308, 9785, 1), (4309, 4317, 1), (4309, 4330, 4294), (4311, 10624, 1), (4316, 4324, 1), (4316, 4330, 4325), (4317, 4325, 4293), (4319, 4320, 1), (4324, 4325, 4330), (4325, 4330, 1), (4345, 6049, 1), (4348, 7221, 1), (4351, 4354, 1), (4652, 6734, 5744), (5004, 5005, 25), (5013, 7745, 7736), (5046, 6848, 3091), (5046, 6905, 6979), (5054, 5071, 2), (5054, 5076, 3851), (5056, 10303, 2), (5059, 5068, 5073), (5068, 11541, 3543), (5073, 8703, 3090), (5077, 7866, 8357), (5085, 5480, 3618), (5128, 5727, 4848), (5175, 5744, 6734), (5177, 6888, 5056), (5189, 7492, 2), (5218, 5229, 12), (5225, 7288, 11), (5261, 5281, 3085), (5265, 5274, 3086), (5318, 11480, 11488), (5319, 7765, 5286), (5321, 11481, 11489), (5432, 10895, 10588), (5433, 10896, 10589), (5446, 9730, 3567), (5447, 5891, 7999), (5550, 9779, 8227), (5550, 10248, 9779), (5584, 6253, 2550), (5587, 6684, 9780), (5806, 11227, 5439), (5878, 6759, 5656), (5881, 11362, 8), (5882, 7982, 3241), (6143, 6639, 2), (6143, 7552, 6639), (6225, 11206, 1498), (6459, 6460, 6), (6560, 9541, 7585), (6636, 7391, 2), (6643, 7387, 3089), (6644, 7517, 3518), (6676, 8889, 2), (6756, 7395, 6997), (6781, 7756, 32), (6803, 7395, 2), (6815, 7503, 2), (6824, 6889, 2), (6824, 6917, 6843), (6825, 6833, 2), (6827, 6848, 6919), (6827, 6905, 2), (6827, 6970, 6963), (6827, 6985, 6848), (6830, 6842, 5141), (6830, 6954, 2), (6831, 6907, 2476), (6831, 6988, 2), (6832, 6989, 2), (6833, 6923, 6871), (6834, 6891, 2), (6834, 6928, 5187), (6835, 6986, 2), (6836, 6925, 4), (6836, 6962, 6943), (6836, 7580, 6838), (6837, 6839, 3091), (6838, 6840, 3091), (6838, 6890, 2), (6842, 6977, 2), (6843, 6870, 3091), (6844, 6871, 3091), (6845, 6937, 5), (6846, 6894, 3091), (6847, 6850, 5177), (6847, 6908, 2), (6848, 6926, 2), (6850, 6851, 4), (6850, 6892, 6937), (6850, 6906, 2), (6853, 6862, 2), (6854, 6883, 2), (6856, 7483, 2), (6863, 6952, 2), (6868, 6869, 4), (6868, 6885, 6936), (6868, 6934, 2), (6868, 6948, 3), (6869, 6938, 3146), (6869, 6948, 6934), (6872, 6904, 5129), (6875, 6917, 2), (6875, 6951, 6889), (6876, 6950, 3), (6878, 6881, 2), (6880, 6882, 2), (6882, 6941, 5154), (6884, 6993, 5056), (6885, 6930, 5), (6885, 6936, 2), (6892, 6937, 2), (6893, 6911, 6964), (6893, 6940, 2), (6895, 6908, 3091), (6899, 6905, 6926), (6899, 6985, 2), (6902, 6924, 2), (6903, 6942, 6891), (6904, 6992, 10303), (6905, 6963, 6970), (6905, 7491, 5046), (6906, 6937, 6892), (6907, 6935, 2), (6909, 6925, 2), (6909, 6932, 6966), (6911, 6947, 2), (6912, 6986, 405), (6914, 6917, 6824), (6914, 6951, 2), (6916, 10431, 6839), (6919, 6979, 5056), (6922, 6927, 2), (6923, 6950, 2), (6924, 6929, 6944), (6925, 6966, 6932), (6928, 6942, 2), (6930, 6948, 6955), (6930, 6955, 2), (6932, 6943, 5), (6932, 6966, 2), (6934, 6936, 6885), (6934, 6938, 4), (6934, 6987, 6904), (6935, 6988, 6910), (6936, 6955, 631), (6938, 6948, 2), (6938, 6955, 6930), (6941, 6961, 2), (6943, 6962, 2), (6944, 6967, 2), (6949, 6958, 2), (6960, 6972, 2), (6962, 6966, 631), (6963, 6970, 2), (6985, 7491, 4), (6997, 7485, 2), (7381, 11340, 2), (7383, 7404, 2), (7388, 11291, 2), (7389, 11292, 2), (7390, 7406, 3091), (7391, 7400, 3091), (7392, 7484, 2), (7395, 7408, 3091), (7396, 7488, 10303), (7396, 10565, 2), (7399, 7409, 3091), (7401, 7509, 2), (7411, 10431, 2), (7464, 7512, 3520), (7470, 11676, 3), (7484, 11403, 11479), (7502, 11250, 3), (7509, 7553, 7394), (7509, 7576, 7401), (7511, 7549, 7557), (7538, 7560, 7520), (7540, 7550, 7533), (7553, 9818, 4), (7555, 7574, 7552), (7576, 9818, 7394), (7689, 11750, 11457), (7714, 10691, 2), (7761, 7816, 7795), (7783, 7823, 7774), (8226, 8728, 6991), (8227, 10165, 5550), (8357, 8369, 7866), (8550, 11477, 1992), (8703, 11541, 5068), (9778, 10430, 63), (9825, 11479, 7392), (10267, 10532, 10587), (10269, 10531, 10586), (10304, 10565, 10298), (10310, 11500, 100), (11015, 11220, 944), (11111, 11112, 2), (11291, 11292, 7819), (11293, 11294, 2), (11413, 11414, 7488)

Notes

As a point on the Euler line, X₂₀ has Shinagawa coefficients \((1, -2)\).

Let L_a be the polar of X₄ wrt the circle centered at A and passing through X₃, and define L_b and L_c cyclically. (Note: X₄ is the perspector of any circle centered at a vertex of ABC.) Let A″ = L_b∩\ L_c, and define B″ and C″ cyclically. Triangle A″B″C″ is homothetic to the anticomplementary triangle, and the center of homothety is X₂₀, which is also the orthocenter of A″B″C″. Also, let L_a be the line through the intersections of the B- and C-Soddy ellipses, and define L_b and L_c cyclically. Then L_a,L_b,L_c concur in X₂₀. Also, let A'B'C' be the cevian triangle of X₂₅₃. Let A″ be the orthocenter of AB'C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₂₀. (Randy Hutson, November 18, 2015)

Let L be the Brocard axis of the intouch triangle. Let L_a be the Brocard axis of the A-extouch triangle, and define L_b and L_c cyclically. The lines L, L_a, L_b, L_c concur in X₂₀. (Randy Hutson, September 14, 2016)

Let A' be the reflection in BC of the A-vertex of the anticevian triangle of X₄, and define B' and C' cyclically. The circumcircles of A'BC, B'CA, and C'AB concur at X₂₀. (Randy Hutson, December 10, 2016)

Let A'B'C' be the reflection of ABC in X₃ (i.e., the circumcevian triangle of X₃). Let A″ be the cevapoint of B' and C', and define B″ and C″ cyclically. The lines AA″, BB″, and CC″ concur in X₂₀. (Randy Hutson, December 10, 2016)

Let A'B'C' be the hexyl triangle. Let A″ be the cevapoint of B' and C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₂₀.

Let A'B'C' be the half-altitude triangle. Let A″ be the trilinear pole of line B'C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₂₀.

Let A'B'C' be the hexyl triangle and A″B″C″ be the side-triangle of ABC and hexyl triangle. Let A* be the {B',C'}-harmonic conjugate of A″, and define B*, C* cyclically. The lines AA*, BB*, CC* concur in X₂₀. (Randy Hutson, June 27, 2018)

Let O_A be the circle centered at the A-vertex of the Ara triangle and passing through A; define O_B and O_C cyclically. X₂₀ is the radical center of O_A, O_B, O_C. (Randy Hutson, August 30, 2020)

Let O_A be the circle centered at the A-vertex of the Johnson triangle and passing through A; define O_B, O_C cyclically. X₂₀ is the radical center of O_A and O_B, O_C. (Randy Hutson, August 30, 2020)

Consider the chain of (blue) right triangles T₁, T₂, T₃, … in this figure: “Pythagorean snail”. The points X₂₀-of-T₁, X₂₀-of-T₂, X(20-of-T₃, … all lie on a single circle. The same sort of concyclicity hold for X_i for these i: 20, 22, 110, 175, 253, 280, 347, 858, 401, 925* 1114, 1294, 1295, 1297, 1303* 1305*, 1370*,… where (*) menas that the point is on theline at infinity. The lines X₃X₆₅₂ of the triangles converge to the center of the snail; it appears that, with repect to each of the triangles, X₆₂₅ lies on the hypotenuse. (Dan Reznik, October 30, 2021)

Let E_a be the ellipse through A having foci B and C, and define E_b and E_c cyclically. These ellipses meet in 6 points. The lines through pairs of opposite intersections concur on X₂₀. Figure. (Dan Reznik, December 12, 2021)

In the plane of a triangle ABC, let

• A' = A-excircle of ABFC, and define B' and C' cyclically; A₁ = orthopole of BC wrt B'C', and define B₁ and C₁ cyclically; Then X₂ = A₁B₁C₁-to-ABC orthology center, and X₃₀₆₂ = ABC-to-A₁B₁C₁ orthology center. (Ivan Pavlov, August 21, 2022)