X₄

ORTHOCENTER

X₄ is the point of concurrence of the altitudes of ABC.

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

X₄ (ORTHOCENTER) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Orthic Triangle △H_aH_bH_c — vertices H_a, H_b, H_c (Pedal triangle of Orthocenter X₄ (altitude feet))

• Concurrence Lines: A–H_a, B–H_b, C–H_c concurring at X₄

• Central Lines: {2, 3} (Euler line)

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

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

Barycentrics: \(1/S_{A} : 1/S_{B} : 1/S_{C}\)

Tripolars: \(|\cos A| : :\)

Search Value (6, 9, 13): \(-5.676619410927\)

▸ Alternate Coordinates9 forms

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

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

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

• Trilinears: \(\csc A \tan 3A - 2 \sec 3A : :\)

• Trilinears: \(4 \cos A - \cos(B - C) - 3 \sin B \sin C : :\)

• Trilinears: \(\cos A + \cos(B - C) + 5 \cos B \cos C - 2 \sin B \sin C : :\)

• Barycentrics: \(\tan A : \tan B : \tan C\)

• Barycentrics: \(1/(b^{2} + c^{2} - a^{2}) : 1/(c^{2} + a^{2} - b^{2}) : 1/(a^{2} + b^{2} - c^{2})\)

• Tripolars: \(|a(b^{2} + c^{2} - a^{2})| : :\)

▸ Linear Combinations2 combinations

• X₄ = \((1 + J)\) X₁₁₁₃ + \((1 - J)\) X₁₁₁₄

• X₄ = \((\tan A)\) A + \((\tan B)\) B + \((\tan C)\) C, where A, B, C are the angles and A, B, C are the vertices

X₄ lies on the Thomson, Darboux, Napoleon, Lucas, McCay, and Neuberg cubics, and the Darboux septic.

Related Collection 1 Topic Guides

T_1: Isogonal Conjugate • T_2: Isotomic Conjugate • T_3: Complement & Anticomplement • T_4: Harmonic Conjugate & Cross-Ratio • T_5: Circle Inversion • T_6: The Euler Line • T_11: The Soddy Line • T_12: The Fuhrmann & Spieker Central Lines • T_13: The Isogonic & Napoleon Central Lines • T_15: The Circumcircle • T_16: The Nine-Point (Feuerbach) Circle • T_17: The Spieker Circle • T_19: The Brocard & Orthocentroidal Circles • T_20: Medial & Anticomplementary Triangles • T_21: Orthic & Tangential Triangles • T_23: Incentral & Excentral Triangles • T_24: Equilateral, Isogonic & Napoleon Triangles

▸ Incident Central Lines953 lines

{1, 33}, {2, 3}, {6, 53}, {7, 273}, {8, 72}, {9, 10}, {11, 56}, {12, 55}, {13, 61}, {14, 62}, {15, 17}, {16, 18}, {31, 3072}, {32, 98}, {35, 498}, {36, 499}, {37, 1841}, {39, 232}, {41, 2202}, {42, 1860}, {46, 90}, {48, 1881}, {49, 156}, {50, 9220}, {51, 185}, {52, 68}, {54, 184}, {57, 84}, {58, 5292}, {63, 5709}, {64, 459}, {65, 158}, {66, 9969}, {67, 338}, {69, 76}, {74, 107}, {75, 12689}, {78, 908}, {79, 1784}, {80, 1825}, {81, 5707}, {83, 182}, {85, 4872}, {93, 562}, {94, 143}, {95, 8797}, {96, 231}, {99, 114}, {100, 119}, {101, 118}, {102, 124}, {103, 116}, {105, 5511}, {106, 5510}, {109, 117}, {110, 113}, {111, 1560}, {120, 1292}, {121, 1293}, {122, 1294}, {123, 1295}, {126, 1296}, {127, 1289}, {128, 930}, {129, 1303}, {130, 1298}, {131, 135}, {137, 933}, {141, 1350}, {142, 5732}, {144, 2894}, {145, 149}, {147, 148}, {150, 152}, {151, 2818}, {154, 8888}, {155, 254}, {157, 5593}, {160, 3613}, {162, 270}, {165, 1698}, {171, 601}, {175, 10905}, {176, 10904}, {177, 8095}, {181, 9553}, {183, 3785}, {187, 7607}, {189, 5908}, {191, 2949}, {193, 1351}, {195, 399}, {200, 6769}, {201, 7069}, {204, 1453}, {210, 7957}, {212, 3074}, {214, 12119}, {215, 9652}, {216, 8799}, {218, 294}, {230, 3053}, {233, 10979}, {236, 8128}, {238, 602}, {240, 256}, {250, 1553}, {251, 8879}, {252, 1487}, {255, 1935}, {276, 327}, {279, 1565}, {280, 2968}, {282, 3345}, {284, 5747}, {290, 6528}, {298, 5864}, {299, 5865}, {312, 7270}, {325, 1975}, {333, 5788}, {339, 10749}, {341, 12397}, {345, 7283}, {346, 3695}, {347, 6356}, {348, 5088}, {354, 3296}, {371, 485}, {372, 486}, {373, 11465}, {385, 7823}, {386, 2051}, {390, 495}, {391, 2322}, {394, 1217}, {477, 1304}, {484, 3460}, {487, 489}, {488, 490}, {493, 8212}, {494, 8213}, {496, 999}, {512, 879}, {518, 6601}, {519, 3680}, {523, 1552}, {524, 5485}, {525, 2435}, {527, 5735}, {528, 3913}, {529, 3813}, {532, 5862}, {533, 5863}, {535, 8666}, {538, 7758}, {539, 9936}, {541, 9140}, {542, 576}, {543, 5503}, {544, 10710}, {551, 9624}, {566, 9221}, {567, 7578}, {569, 1179}, {572, 1474}, {574, 1506}, {575, 598}, {579, 1713}, {580, 1714}, {584, 8818}, {590, 1151}, {595, 8750}, {603, 3075}, {604, 7120}, {608, 1518}, {615, 1152}, {616, 627}, {617, 628}, {618, 5473}, {619, 5474}, {620, 7862}, {625, 3788}, {626, 3734}, {635, 3643}, {636, 3642}, {639, 5590}, {640, 5591}, {641, 12124}, {642, 12123}, {651, 3157}, {653, 1156}, {674, 12587}, {684, 2797}, {685, 2682}, {690, 11005}, {693, 8760}, {695, 3981}, {754, 7751}, {758, 5693}, {774, 1254}, {800, 13380}, {801, 1092}, {842, 935}, {885, 3309}, {912, 3868}, {916, 2997}, {936, 3452}, {937, 1534}, {940, 1396}, {941, 1880}, {953, 1309}, {958, 2886}, {960, 5794}, {970, 9534}, {972, 5514}, {973, 6145}, {974, 7729}, {983, 5255}, {990, 4000}, {991, 4648}, {993, 11012}, {1000, 3057}, {1015, 9651}, {1029, 2906}, {1032, 5910}, {1034, 5911}, {1036, 1065}, {1037, 1067}, {1038, 1076}, {1039, 1096}, {1040, 1074}, {1041, 2263}, {1043, 4417}, {1046, 2648}, {1060, 4296}, {1062, 3100}, {1073, 2130}, {1078, 5171}, {1089, 3974}, {1104, 3772}, {1111, 4056}, {1123, 7133}, {1125, 3576}, {1131, 3311}, {1132, 3312}, {1138, 2132}, {1139, 3368}, {1140, 3395}, {1157, 2120}, {1160, 1162}, {1161, 1163}, {1164, 3595}, {1165, 3593}, {1175, 5320}, {1177, 5622}, {1192, 3532}, {1209, 4549}, {1216, 2979}, {1248, 2660}, {1251, 1832}, {1260, 5687}, {1317, 12763}, {1319, 7704}, {1327, 6419}, {1328, 6420}, {1329, 1376}, {1336, 2362}, {1340, 1348}, {1341, 1349}, {1342, 1676}, {1343, 1677}, {1353, 5093}, {1379, 2040}, {1380, 2039}, {1383, 8791}, {1384, 8778}, {1385, 1538}, {1389, 2099}, {1392, 3241}, {1393, 7004}, {1399, 5348}, {1420, 4311}, {1430, 1468}, {1435, 3333}, {1440, 7053}, {1441, 4329}, {1445, 3358}, {1448, 7365}, {1469, 12589}, {1483, 3623}, {1484, 12773}, {1495, 3431}, {1499, 1550}, {1500, 9650}, {1510, 10412}, {1511, 12121}, {1521, 7115}, {1562, 6529}, {1566, 2724}, {1609, 9722}, {1621, 10267}, {1670, 5404}, {1671, 5403}, {1682, 9552}, {1689, 2010}, {1690, 2009}, {1691, 3406}, {1697, 7160}, {1715, 1730}, {1716, 1721}, {1717, 1718}, {1726, 1782}, {1729, 8558}, {1764, 10479}, {1768, 3065}, {1773, 2961}, {1781, 2955}, {1798, 13323}, {1840, 4876}, {1903, 2262}, {1942, 2790}, {1957, 5247}, {1970, 1971}, {1973, 2201}, {1987, 3269}, {1994, 2904}, {2077, 3814}, {2080, 7793}, {2092, 3597}, {2093, 4848}, {2095, 9965}, {2098, 10944}, {2121, 3481}, {2131, 3349}, {2133, 8440}, {2181, 4642}, {2217, 3417}, {2275, 9597}, {2276, 9596}, {2278, 5397}, {2287, 5778}, {2331, 3755}, {2332, 4251}, {2353, 3425}, {2355, 3579}, {2361, 7299}, {2393, 5486}, {2456, 10349}, {2457, 3667}, {2477, 9653}, {2482, 12117}, {2536, 2540}, {2537, 2541}, {2574, 2592}, {2575, 2593}, {2646, 4305}, {2651, 2907}, {2679, 2698}, {2687, 2766}, {2697, 10423}, {2734, 10017}, {2752, 10101}, {2770, 10098}, {2771, 9803}, {2778, 10693}, {2783, 10769}, {2784, 11599}, {2787, 10768}, {2791, 4516}, {2793, 9180}, {2801, 3254}, {2802, 12641}, {2814, 3762}, {2817, 13532}, {2822, 4466}, {2823, 4858}, {2826, 10773}, {2827, 10774}, {2828, 10775}, {2830, 10779}, {2831, 10780}, {2840, 4939}, {2889, 6101}, {2896, 6194}, {2900, 3189}, {2905, 6625}, {2908, 7139}, {2917, 8146}, {2929, 2935}, {2972, 10745}, {2975, 5841}, {2995, 8048}, {3023, 12185}, {3024, 12374}, {3027, 12184}, {3028, 12373}, {3054, 5210}, {3056, 12588}, {3058, 3303}, {3062, 3339}, {3094, 3399}, {3096, 3098}, {3101, 8251}, {3120, 3924}, {3162, 5359}, {3164, 9290}, {3172, 3424}, {3180, 5873}, {3181, 5872}, {3184, 6716}, {3190, 3191}, {3212, 7261}, {3216, 5400}, {3218, 5770}, {3255, 5883}, {3270, 11461}, {3304, 5434}, {3305, 3587}, {3306, 7171}, {3314, 7885}, {3320, 12945}, {3329, 7864}, {3338, 7284}, {3340, 3577}, {3342, 3347}, {3344, 3348}, {3352, 3354}, {3356, 3637}, {3364, 3391}, {3365, 3392}, {3366, 3389}, {3367, 3390}, {3369, 3397}, {3370, 3396}, {3371, 3387}, {3372, 3388}, {3373, 3385}, {3374, 3386}, {3379, 5402}, {3380, 5401}, {3381, 3394}, {3382, 3393}, {3398, 3407}, {3413, 3558}, {3414, 3557}, {3416, 3714}, {3426, 13093}, {3430, 3454}, {3438, 3443}, {3439, 3442}, {3440, 5682}, {3441, 5681}, {3461, 7165}, {3463, 5683}, {3466, 3469}, {3479, 3489}, {3480, 3490}, {3495, 8866}, {3497, 7351}, {3499, 8925}, {3500, 7350}, {3502, 8867}, {3521, 5946}, {3527, 8796}, {3580, 11472}, {3582, 4325}, {3584, 4330}, {3589, 5085}, {3590, 6221}, {3591, 6398}, {3601, 4304}, {3611, 11460}, {3614, 5217}, {3617, 5690}, {3620, 7879}, {3621, 5844}, {3622, 5901}, {3624, 7987}, {3629, 5102}, {3632, 4900}, {3633, 11224}, {3634, 10164}, {3648, 3652}, {3668, 8809}, {3671, 5665}, {3679, 4866}, {3701, 5300}, {3704, 5695}, {3706, 10371}, {3738, 10771}, {3741, 10476}, {3746, 4309}, {3753, 9800}, {3812, 5880}, {3815, 5013}, {3819, 13348}, {3820, 6244}, {3822, 5248}, {3825, 10200}, {3826, 11495}, {3829, 11194}, {3841, 7688}, {3847, 6691}, {3849, 7615}, {3870, 5534}, {3871, 10528}, {3877, 7700}, {3885, 12648}, {3887, 10772}, {3911, 6705}, {3916, 5744}, {3917, 7999}, {3925, 5584}, {3933, 7776}, {3934, 5188}, {3940, 5763}, {3947, 4314}, {3972, 7828}, {4008, 12723}, {4045, 7808}, {4048, 5103}, {4277, 4646}, {4308, 7743}, {4313, 5226}, {4316, 7280}, {4317, 5563}, {4324, 5010}, {4339, 5266}, {4355, 10980}, {4357, 10444}, {4423, 7958}, {4425, 8235}, {4444, 6002}, {4512, 10268}, {4645, 7155}, {4654, 11518}, {4658, 5733}, {4692, 4894}, {4721, 4805}, {4723, 12693}, {4768, 9525}, {4846, 5462}, {4847, 12527}, {4863, 12692}, {5007, 5309}, {5008, 5346}, {5032, 11405}, {5038, 11170}, {5044, 10157}, {5045, 5558}, {5050, 5395}, {5092, 7859}, {5097, 7894}, {5119, 7162}, {5121, 11512}, {5123, 13528}, {5173, 12677}, {5204, 5433}, {5206, 6781}, {5221, 10308}, {5223, 12777}, {5249, 10884}, {5253, 10269}, {5265, 10593}, {5273, 5791}, {5278, 9958}, {5281, 10592}, {5377, 6074}, {5418, 6200}, {5420, 6396}, {5424, 5441}, {5435, 5704}, {5437, 9841}, {5438, 6700}, {5439, 9776}, {5440, 5748}, {5447, 7998}, {5449, 7689}, {5461, 10153}, {5505, 10752}, {5513, 9085}, {5533, 10074}, {5535, 6597}, {5536, 6763}, {5542, 6744}, {5550, 11230}, {5553, 7702}, {5556, 10977}, {5557, 12005}, {5559, 5697}, {5561, 11552}, {5597, 8196}, {5598, 8203}, {5599, 11822}, {5600, 11823}, {5601, 8200}, {5602, 8207}, {5606, 5950}, {5609, 5655}, {5623, 8446}, {5624, 8456}, {5627, 6070}, {5670, 8487}, {5671, 8494}, {5672, 8444}, {5673, 8454}, {5674, 8495}, {5675, 8496}, {5676, 8486}, {5677, 7329}, {5678, 8491}, {5679, 8492}, {5680, 7164}, {5685, 8480}, {5688, 12698}, {5689, 12697}, {5705, 5745}, {5708, 12684}, {5795, 9623}, {5848, 10759}, {5853, 6765}, {5854, 13271}, {5860, 6278}, {5861, 6281}, {5874, 11917}, {5875, 11916}, {5885, 10266}, {5892, 11451}, {5933, 10362}, {5934, 8079}, {5935, 7593}, {5943, 9729}, {5951, 5952}, {5965, 7877}, {5984, 7766}, {6020, 12955}, {6032, 12506}, {6036, 7857}, {6055, 9166}, {6073, 11607}, {6075, 10428}, {6082, 6092}, {6114, 9750}, {6115, 9749}, {6128, 8749}, {6130, 9409}, {6147, 11036}, {6196, 8927}, {6204, 8957}, {6217, 6266}, {6218, 6267}, {6219, 6276}, {6220, 6277}, {6224, 6265}, {6233, 13234}, {6235, 8705}, {6238, 10055}, {6285, 7049}, {6292, 7935}, {6323, 12494}, {6326, 6596}, {6339, 10981}, {6407, 9542}, {6409, 8253}, {6410, 8252}, {6453, 9681}, {6462, 8220}, {6463, 8221}, {6467, 12283}, {6519, 9692}, {6680, 7844}, {6704, 9751}, {6735, 12534}, {6752, 8795}, {6777, 11602}, {6778, 11603}, {7017, 7141}, {7028, 8127}, {7059, 7345}, {7060, 7344}, {7149, 8811}, {7161, 11010}, {7264, 7272}, {7320, 9785}, {7325, 8449}, {7326, 8459}, {7327, 8432}, {7352, 10071}, {7587, 8379}, {7588, 8086}, {7589, 8382}, {7595, 12681}, {7603, 11669}, {7605, 13339}, {7617, 8182}, {7618, 8176}, {7666, 10272}, {7676, 7679}, {7677, 7678}, {7693, 13363}, {7703, 11454}, {7712, 10610}, {7720, 7725}, {7721, 7726}, {7723, 12219}, {7730, 7731}, {7739, 7753}, {7757, 7858}, {7769, 7782}, {7777, 7783}, {7778, 7789}, {7779, 7900}, {7786, 7847}, {7792, 7851}, {7794, 7818}, {7796, 7809}, {7798, 7838}, {7799, 7814}, {7801, 7821}, {7804, 7834}, {7813, 7903}, {7815, 7830}, {7820, 7867}, {7822, 7853}, {7829, 7902}, {7831, 7910}, {7832, 7934}, {7835, 7899}, {7836, 7912}, {7839, 7921}, {7845, 7855}, {7846, 7919}, {7854, 7873}, {7856, 12150}, {7863, 7888}, {7875, 7923}, {7883, 10302}, {7889, 7913}, {7891, 7925}, {7905, 7926}, {7906, 7941}, {7932, 10583}, {8068, 10058}, {8069, 10321}, {8075, 8087}, {8076, 8088}, {8077, 8085}, {8080, 8092}, {8099, 9793}, {8100, 9795}, {8105, 8426}, {8106, 8427}, {8107, 8380}, {8108, 8381}, {8109, 8377}, {8110, 8378}, {8117, 8123}, {8118, 8124}, {8125, 8129}, {8126, 8130}, {8141, 9536}, {8144, 9538}, {8172, 8447}, {8173, 8457}, {8193, 9911}, {8197, 12458}, {8204, 12459}, {8222, 11828}, {8223, 11829}, {8224, 8230}, {8225, 8228}, {8372, 12674}, {8431, 8443}, {8433, 8483}, {8434, 8484}, {8435, 8481}, {8436, 8482}, {8437, 8497}, {8438, 8498}, {8445, 8458}, {8448, 8455}, {8450, 8461}, {8451, 8460}, {8452, 8463}, {8453, 8462}, {8488, 8527}, {8489, 8532}, {8490, 8533}, {8501, 8509}, {8502, 8508}, {8515, 8536}, {8516, 8535}, {8517, 8534}, {8538, 11416}, {8582, 10860}, {8583, 10863}, {8588, 10185}, {8591, 8724}, {8596, 12355}, {8674, 10767}, {8679, 12586}, {8719, 10155}, {8864, 8921}, {8868, 8872}, {8878, 10340}, {8983, 9583}, {9147, 11615}, {9300, 9607}, {9530, 10718}, {9627, 9629}, {9628, 9630}, {9638, 10535}, {9646, 9660}, {9647, 9661}, {9648, 9662}, {9649, 9663}, {9658, 9672}, {9659, 9673}, {9705, 13482}, {9783, 12488}, {9787, 12489}, {9789, 12490}, {9791, 9959}, {9845, 12577}, {9857, 12497}, {9874, 12139}, {9897, 11280}, {9919, 13171}, {9934, 13198}, {9942, 10391}, {9967, 12220}, {9973, 13622}, {10042, 10050}, {10043, 10057}, {10052, 10073}, {10088, 12896}, {10187, 10646}, {10188, 10645}, {10202, 11220}, {10264, 10620}, {10293, 12099}, {10305, 11023}, {10309, 12676}, {10313, 10316}, {10363, 10369}, {10415, 10422}, {10434, 10887}, {10435, 12547}, {10455, 10464}, {10529, 10680}, {10546, 10564}, {10547, 10548}, {10627, 13340}, {10634, 11420}, {10635, 11421}, {10707, 11240}, {10791, 12197}, {10797, 10799}, {10798, 12835}, {10831, 10833}, {10873, 10877}, {10882, 10886}, {10897, 11417}, {10898, 11418}, {10912, 13463}, {10915, 12703}, {10916, 12704}, {10923, 10927}, {10924, 10928}, {10956, 10965}, {10957, 10966}, {10958, 11502}, {11082, 11135}, {11087, 11136}, {11171, 11272}, {11177, 11632}, {11270, 11468}, {11402, 11426}, {11408, 11485}, {11409, 11486}, {11423, 13366}, {11449, 12038}, {11557, 11560}, {11587, 13558}, {11646, 13330}, {11649, 11663}, {11698, 12331}, {11703, 12165}, {11755, 11759}, {11764, 11768}, {11773, 11777}, {11782, 11786}, {11792, 13508}, {11800, 12284}, {11869, 11873}, {11870, 11874}, {11891, 12491}, {11900, 12696}, {11905, 11909}, {11930, 11947}, {11931, 11948}, {11990, 11992}, {12006, 13364}, {12061, 12063}, {12120, 12864}, {12146, 12849}, {12166, 12309}, {12168, 12310}, {12169, 12311}, {12170, 12312}, {12171, 12313}, {12172, 12314}, {12175, 12316}, {12223, 12603}, {12224, 12604}, {12226, 12606}, {12271, 12272}, {12273, 12280}, {12350, 12354}, {12369, 13495}, {12387, 12394}, {12388, 12393}, {12507, 13249}, {12515, 12619}, {12516, 12620}, {12517, 12621}, {12518, 12622}, {12519, 12623}, {12520, 12609}, {12521, 12612}, {12522, 12613}, {12523, 12614}, {12524, 12615}, {12556, 13089}, {12624, 13238}, {12739, 12743}, {12837, 13077}, {12859, 12863}, {12941, 13075}, {12942, 13076}, {12944, 13078}, {12946, 13079}, {12947, 13080}, {12948, 13081}, {12949, 13082}, {13007, 13023}, {13008, 13024}, {13009, 13039}, {13010, 13040}, {13321, 13451}, {13353, 13470}, {13418, 13423}

▸ Functional Relations120 relations

• X₄ = midpoint of X_i and X_j for these (i,j): (3, 382), (146, 3448), (147, 148), (149, 153), (150, 152)

• X₄ = reflection of X_i in X_j for these (i,j): (1, 946), (2, 381), (3, 5), (5, 546), (8, 355), (20, 3), (24, 235), (25, 1596), (40, 10), (69, 1352), (74, 125), (98, 115), (99, 114), (100, 119), (101, 118), (102, 124), (103, 116), (104, 11), (107, 133), (109, 117), (110, 113), (112, 132), (145, 1482), (185, 389), (186, 403), (193, 1351), (376, 2), (378, 427), (550, 140), (917, 5190), (925, 131), (930, 128), (944, 1), (1113, 1312), (1114, 1313), (1141, 137), (1292, 120), (1293, 121), (1294, 122), (1295, 123), (1296, 126), (1297, 127), (1298, 130), (1299, 135), (1300, 136), (1303, 129), (1350, 141), (1593, 1595)

• X₄ = isogonal conjugate of X₃

• X₄ = isotomic conjugate of X₆₉

• X₄ = cyclocevian conjugate of X₂

• X₄ = circumcircle-inverse of X₁₈₆

• X₄ = nine-point-circle-inverse of X₄₀₃

• X₄ = complement of X₂₀

• X₄ = anticomplement of X₃

• X₄ = complementary conjugate of X₂₈₈₃

• X₄ = anticomplementary conjugate of X₂₀

• X₄ = 2nd-Brocard-circle-inverse of X₃₇₉₉₁

• X₄ = Grebe-circle-inverse of X₃₇₉₂₅

• X₄ = X_i-Ceva conjugate of X_j for these (i,j): (2, 1249), (7, 196), (27, 19), (29, 1), (92, 281), (107, 523), (264, 2), (273, 278), (275, 6), (286, 92), (393, 459)

• X₄ = cevapoint of X_i and X_j for these (i,j): (1, 46), (2, 193), (3, 155), (5, 52), (6, 25), (11, 513), (19, 33), (30, 113), (34, 208), (37, 209), (39, 211), (51, 53), (65, 225), (114, 511), (115, 512), (116, 514), (117, 515), (118, 516), (119, 517), (120, 518), (121, 519), (122, 520), (123, 521), (124, 522), (125, 523), (126, 524), (127, 525), (185, 235), (371, 372), (487, 488)

• X₄ = X_i-cross conjugate of X_j for these (i,j): (3, 254), (6, 2), (19, 278), (25, 393), (33, 281), (51, 6), (52, 24), (65, 1), (113, 403), (125, 523), (185, 3), (193, 459), (225, 158), (389, 54), (397, 17), (398, 18), (407, 225), (427, 264), (512, 112), (513, 108), (523, 107)

• X₄ = crosspoint of X_i and X_j for these (i,j): (2, 253), (7, 189), (27, 286), (92, 273)

• X₄ = crosssum of X_i and X_j for these (i,j): (4, 1075), (6, 154), (25, 1033), (48, 212), (55, 198), (56, 1035), (71, 228), (184, 577), (185, 417), (216, 418)

• X₄ = crossdifference of every pair of points on line X₅₂₀X₆₄₇

• X₄ = X_i-Hirst inverse of X_j for these (i,j): (1, 243), (2, 297), (3, 350), (19, 242), (21, 425), (24, 421), (25, 419), (27, 423), (28, 422), (29, 415), (420, 427), (424, 451), (459, 460), (470, 471), (1249, 1503)

• X₄ = X_i-aleph conjugate of X_j for these (i,j): (1, 1047), (29, 4)

• X₄ = X_i-beth conjugate of X_j for these (i,j): (4, 34), (8, 40), (29, 4), (162, 56), (318, 318), (811, 331)

• X₄ = intersection of tangents at X₃ and X₄ to McCay cubic K003

• X₄ = intersection of tangents at X₄ and X₆₉ to Lucas cubic K007

• X₄ = exsimilicenter of 1st & 2nd Johnson-Yff circles; the insimilicenter is X₁

• X₄ = trilinear pole of PU(4) (the orthic axis)

• X₄ = trilinear pole wrt orthic triangle of orthic axis

• X₄ = trilinear pole wrt intangents triangle of orthic axis

• X₄ = trilinear pole wrt circumsymmedial triangle of orthic axis

• X₄ = trilinear product of PU(15)

• X₄ = barycentric product of PU(i) for these i: 21, 45

• X₄ = bicentric sum of PU(i) for these i: 126, 131

• X₄ = PU(126)-harmonic conjugate of X₆₅₂

• X₄ = midpoint of PU(131)

• X₄ = crosspoint of polar conjugates of PU(4)

• X₄ = cevapoint of foci of orthic inconic

• X₄ = QA-P33 (Centroid of the Orthocenter Quadrangle) of quadrangle ABCX₂ (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/61-qa-p33.html)

• X₄ = Hofstadter -1 point

• X₄ = X₄-of X_iX_jX_k for these {i,j,k}: {1, 8, 5556}, {1, 9, 885}, {2, 6, 1640}, {2, 10, 4049}, {3, 6, 879}, {3, 66, 2435}, {7, 8, 885}

• X₄ = homothetic center of these triangles: orthic, X₁₃-Ehrmann, X₁₄-Ehrmann (see X₂₅)

• X₄ = perspector of anticomplementary circle

• X₄ = pole wrt polar circle of trilinear polar of X₂ (line at infinity)

• X₄ = pole wrt {circumcircle, nine-point circle}-inverter of Lemoine axis

• X₄ = X₄₈-isoconjugate (polar conjugate) of X₂

• X₄ = X_i-isoconjugate of X_j for these (i,j): (6, 63), (75, 184), (91, 1147), (92, 577), (1101, 125), (2962, 49), (2964, 3519)

• X₄ = X₁₃₄₂-vertex conjugate of X₁₃₄₃

• X₄ = Zosma transform of X₁

• X₄ = X₁₃₅₂ of 1st anti-Brocard triangle

• X₄ = centroid of the union of X₈ and its 3 extraversions

• X₄ = X₅ of extraversion triangle of X₈

• X₄ = homothetic center of orthic triangle and reflection of tangential triangle in X₅

• X₄ = homothetic center of 2nd circumperp and 3rd Euler triangles

• X₄ = trilinear product of vertices of half-altitude triangle

• X₄ = trilinear product of vertices of orthocentroidal triangle

• X₄ = trilinear product of vertices of reflection triangle

• X₄ = trilinear product of vertices of 4th Brocard triangle

• X₄ = center of conic that is the locus of orthopoles of lines passing through X₄

• X₄ = perspector of circumanticevian triangle of X₄ and unary cofactor triangle of circumanticevian triangle of X₃

• X₄ = X₃-of-2nd-extouch-triangle

• X₄ = perspector of ABC and 2nd and 3rd extouch triangles

• X₄ = perspector of ABC and 1st Brocard triangle of anticomplementary triangle

• X₄ = perspector of ABC and 1st Brocard triangle of Johnson triangle

• X₄ = perspector of ABC and mid-triangle of 2nd and 3rd extouch triangles

• X₄ = perspector of extouch triangle and cross-triangle of ABC and 2nd extouch triangle

• X₄ = perspector of 2nd Hyacinth triangle and cross-triangle of ABC and 2nd Hyacinth triangle

• X₄ = homothetic center of 2nd Johnson-Yff triangle and cross-triangle of ABC and 1st Johnson-Yff triangle

• X₄ = homothetic center of 1st Johnson-Yff triangle and cross-triangle of ABC and 2nd Johnson-Yff triangle

• X₄ = X₁-of-orthic-triangle if ABC is acute, and an excenter of orthic triangle otherwise

• X₄ = X₅₂-of-excentral triangle

• X₄ = X₆₅-of-tangential-triangle if ABC is acute

• X₄ = X₁₅₅-of-intouch-triangle

• X₄ = X₁₁₀-of-Fuhrmann-triangle

• X₄ = X₁₄₇-of-1st-Brocard-triangle

• X₄ = X₁₂₉₆-of-4th-Brocard-triangle

• X₄ = X₇₄-of-orthocentroidal-triangle

• X₄ = X₁₁₀-of-X₄-Brocard-triangle

• X₄ = harmonic center of circle O(PU(4)) and orthoptic circle of Steiner inellipse

• X₄ = Thomson-isogonal conjugate of X₁₅₄

• X₄ = Lucas-isogonal conjugate of X₁₁₂₀₆

• X₄ = perspector of ABC and cross-triangle of 1st and 2nd Neuberg triangles

• X₄ = perspector of circumconic centered at X₁₂₄₉

• X₄ = center of circumconic that is locus of trilinear poles of lines passing through X₁₂₄₉

• X₄ = circumcevian isogonal conjugate of X₄

• X₄ = orthic-isogonal conjugate of X₄

• X₄ = X₁-of-circumorthic-triangle if ABC is acute

• X₄ = isogonal conjugate wrt half-altitude triangle of X₁₈₅

• X₄ = Miquel associate of X₄

• X₄ = crosspoint of X₃ and X₁₅₅ wrt both the excentral and tangential triangles

• X₄ = crosspoint of X₄₈₇ and X₄₈₈ wrt both the excentral and anticomplementary triangles

• X₄ = X₃-of-Ehrmann-mid-triangle

• X₄ = X₁₁₀-of-X₃-Fuhrmann-triangle

• X₄ = barycentric product X₁₁₂*X₈₅₀

• X₄ = Kosnita(X₂₀,X₂₀) point

• X₄ = perspector of ABC and the reflection of the excentral triangle in X₁₀

• X₄ = pedal antipodal perspector of X₃

• X₄ = Ehrmann-side-to-Ehrmann-vertex similarity image of X₃

• X₄ = Ehrmann-vertex-to-orthic similarity image of X₄

• X₄ = Ehrmann-side-to-orthic similarity image of X₃

• X₄ = Ehrmann-mid-to-ABC similarity image of X₅

• X₄ = perspector of hexyl triangle and cevian triangle of X₂₇

• X₄ = perspector of hexyl triangle and anticevian triangle of X₁₉

• X₄ = perspector of ABC and medial triangle of pedal triangle of X₆₄

• X₄ = perspector of ABC and the reflection in X₂ of the antipedal triangle of X₂

• X₄ = perspector of hexyl triangle and tangential triangle wrt excentral triangle of the excentral-hexyl ellipse

• X₄ = inverse-in-Steiner-circumellipse of X₂₉₇

• X₄ = {X₂₄₇₉,X₂₄₈₀}-harmonic conjugate of X₂₉₇

• X₄ = symgonal of every point on the nine-point circle

• X₄ = center of bianticevian conic of PU(4) (this conic being the polar circle)

• X₄ = orthoptic-circle-of-Steiner-inellipse inverse of X₄₆₈

• X₄ = de-Longchamps-circle inverse of X₂₀₇₁

• X₄ = center of inverse-in-de-Longchamps-circle of circumcircle

• X₄ = inner-Napoleon circle-inverse of X₃₂₄₆₀

• X₄ = outer-Napoleon circle-inverse of X₃₂₄₆₁

• X₄ = excentral-to-ABC functional image of X₁

• X₄ = orthic-to-ABC functional image of X₅₂

• X₄ = incentral-to-ABC functional image of X₅₀₀

• X₄ = Feuerbach-to-ABC functional image of X₅₉₄₈

• X₄ = excentral-to-intouch similarity image of X₈₄

• X₄ = barycentric product of circumcircle intercepts of line X₂₉₇(525)

• X₄ = trilinear product of vertices of infinite altitude triangle

Notes

As a point on the Euler line, X₄ has Shinagawa coefficients \((0, 1)\).

The tangents at A, B, C to the McCay cubic K003 concur in X₄. Also, the tangents at A, B, C to the Lucas cubic K007 concur in X₄. (Randy Hutson, November 18, 2015)

Let P be a point in the plane of ABC. Let O_a be the circumcenter of BCP, and define O_b and O_c cyclically. Let Q be the circumcenter of O_aO_bO_c. P = Q only when P = X₄. (Randy Hutson, June 27, 2018)

Taking a reference triangle ABC and a variable point P on the plane, P=X₃ is the point of maximal area of its pedal triangle when considering all points P inside the circumcircle of ABC. There are points P far away from the circumcircle for which the area of their pedal triangles is much larger. However, if you consider the signed area of the pedal triangle of P (of which sign depends on whether the points are in clockwise or anti-clockwise order), you could just say that the area of the pedal triangle of P is always negative whenever P is outside of the circumcircle so that P=X₃ gives the global maximum. (Mark Helman, July 10, 2020)

A slightly similar thing happens regarding the area of the antipedal triangle of P. P=X₄ has the smallest area of its antipedal amongst all P in the interior of triangle ABC (when X₄ is in this interior). There are points P (on the circumcircle) for which this area goes to 0. However, if we consider the signed area of the antipedal, even though there are still regions of the plane outside of ABC where the signed area is positive, P=X₄ gives the smallest area of the antipedal among all P for which this area is positive (this works even when ABC is obtuse, and points close to the circumcircle (on both sides) have negative antipedal area). (Mark Helman, July 10, 2020)

View Extremal Area Pedal and Antipedal Triangles, by Mark Helman, Ronaldo Garcia, and Dan Reznik.

X₄ and the vertices A, B, C comprise an orthocentric system, defined as a set of four points, one of which is the orthocenter of the triangle of the other three (so that each is the orthocenter of the other three). The incenter and excenters also form an orthocentric system. If ABC is acute, then X₄ is the incenter of the orthic triangle.

Suppose P is not on a sideline of ABC. Let A'B'C' be the cevian triangle of P. Let D, E, F be the circles having diameters AA', BB',CC'. The radical center of D, E, F is X₄. (Floor van Lamoen, Hyacinthos #214, Jan. 24, 2000.)

Let A₂B₂C₂ be the 2nd Conway triangle. Let A' be the crosspoint of B₂ and C₂, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₄. (Randy Hutson, December 10, 2016)

Let T be any one of these trianges: {anticevian of X₃₀, anti-Hutson-intouch, anti-incircle-circles, Ehrmann side, X₂-Ehrmann, Gemini 15, Gemini 16, Kosnita, midheight, N-obverse of X₆, Schroeter, tangential, Trinh, 1st Zaniah, 2nd Zaniah}. Let O_A be the circle centered at the A-vertex of T and passing through A; define O_B and O_C cyclically. X₄ is the radical center of O_A, O_B, O_C. (Randy Hutson, August 30, 2020)

See Ross Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, Mathematical Association of America, 1995. Chapter 2: The Orthocenter.

In the plane of a triangle ABC, let

• O_a = circle with diameter BC, and define O_b and O_c cyclically;

• A' = reflection of A in O_a, and define B' and C' cyclically;

• A_b = polar of A' with respect to O_b, and define B_c and C_a cyclically;

• A_c = polar of A' with respect to OC, and define B_a and C_b cyclically;

• ″ = A_b∩A_c, and define B″ and C″ cyclically.

The triangle A″B″C″ is perspective to ABC, and the perspector is X₄. (Dasari Naga Vijay Krishna, April 19, 2021)

In the plane of a triangle ABC, let

• O_a = circle with diameter BC, and define O_b and O_c cyclically;

• P_a = polar of A with respect to O_a, and define P_b and P_c cyclically.

Then X₄= P_a∩P_b∩P_c, (Dasari Naga Vijay Krishna, July 19, 2021)

For extensions of triangle geometry to 3-dimensional shapes called orthocentric tetrahedra, see

William Barker and Roger Howe, Continuous Symmetry From Euclid to Klein, American Mathematical Society, 2007, pages 306-309.

X₄ is the {X₃,X₅}-harmonic conjugate of X₂.