X₁₁₀

FOCUS OF KIEPERT PARABOLA

Scimemi proves that if the Euler line is reflected in every side of triangle ABC, then the three reflections concur in X₁₁₀.

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

X₁₁₀ (FOCUS OF KIEPERT PARABOLA) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Circle: Circumcircle (circumcircle) — Passes through vertices A, B, C; centered at X₃ CIRCUMCENTER

• Related Centers: X₃ (CIRCUMCENTER)

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

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

Tripolars: \(|\sin(B - C)| : :\)

Search Value (6, 9, 13): \(0.907562974102\)

▸ Alternate Coordinates4 forms

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

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

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

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

X₁₁₀ lies on the circumcircle, Walsmith rectangular hyperbola, Parry circle.

▸ Incident Central Lines993 lines

{1, 60}, {2, 98}, {3, 74}, {4, 113}, {5, 49}, {6, 111}, {10, 2372}, {11, 215}, {12, 2477}, {13, 3200}, {14, 3201}, {15, 2378}, {16, 2379}, {17, 3205}, {18, 3206}, {20, 146}, {21, 104}, {22, 154}, {23, 323}, {24, 155}, {25, 1112}, {26, 7731}, {27, 917}, {28, 915}, {29, 2988}, {30, 477}, {31, 593}, {32, 729}, {33, 9637}, {35, 501}, {36, 5127}, {39, 755}, {40, 37405}, {41, 35106}, {42, 28482}, {48, 2249}, {50, 32730}, {51, 1994}, {52, 2383}, {55, 3024}, {56, 3028}, {58, 106}, {59, 2222}, {61, 2380}, {62, 2381}, {63, 6061}, {64, 5896}, {65, 229}, {66, 15116}, {67, 141}, {68, 7505}, {69, 206}, {75, 21430}, {76, 2367}, {81, 105}, {83, 3203}, {86, 675}, {97, 418}, {99, 690}, {100, 643}, {101, 163}, {102, 283}, {103, 1790}, {107, 648}, {108, 162}, {109, 1813}, {112, 1625}, {115, 3044}, {116, 3046}, {126, 3048}, {140, 10264}, {143, 195}, {148, 16278}, {149, 19642}, {159, 12220}, {165, 9904}, {183, 9769}, {185, 17701}, {186, 7722}, {187, 352}, {190, 835}, {193, 1974}, {199, 22139}, {213, 2375}, {235, 22750}, {237, 2080}, {248, 9513}, {249, 512}, {250, 520}, {251, 694}, {255, 26701}, {270, 3562}, {274, 767}, {275, 30506}, {284, 2291}, {290, 36901}, {324, 436}, {325, 2857}, {333, 1311}, {343, 10192}, {351, 526}, {353, 574}, {355, 12889}, {371, 7598}, {372, 7599}, {373, 575}, {376, 541}, {378, 11472}, {381, 7699}, {382, 1539}, {384, 35399}, {385, 9418}, {386, 3031}, {389, 12227}, {395, 11134}, {396, 11137}, {402, 13212}, {403, 32123}, {467, 23233}, {468, 3564}, {476, 523}, {489, 30427}, {490, 30428}, {493, 13215}, {494, 13216}, {513, 1290}, {514, 2690}, {515, 2695}, {516, 2688}, {517, 1325}, {518, 2752}, {519, 2758}, {521, 2766}, {522, 2689}, {524, 2770}, {525, 935}, {539, 37943}, {543, 9144}, {546, 37472}, {548, 14677}, {549, 11694}, {550, 8718}, {560, 715}, {568, 12106}, {569, 3090}, {573, 34453}, {576, 9716}, {577, 26717}, {578, 3091}, {588, 8956}, {595, 849}, {599, 6325}, {620, 15357}, {631, 6699}, {632, 37471}, {645, 3952}, {647, 2715}, {649, 2702}, {650, 9090}, {660, 4599}, {663, 2701}, {664, 1305}, {667, 2703}, {668, 839}, {669, 805}, {670, 689}, {671, 5465}, {677, 23090}, {681, 823}, {685, 850}, {693, 2864}, {699, 1501}, {707, 18899}, {719, 1923}, {727, 2206}, {737, 3117}, {739, 1333}, {745, 1964}, {753, 3736}, {758, 12030}, {789, 799}, {813, 1252}, {825, 28860}, {826, 1287}, {827, 4611}, {833, 3888}, {840, 3286}, {841, 7464}, {846, 19655}, {858, 1503}, {859, 953}, {868, 6033}, {880, 4609}, {892, 8599}, {898, 4567}, {901, 1618}, {902, 1326}, {905, 2722}, {906, 36080}, {912, 2074}, {916, 2073}, {919, 5098}, {924, 10420}, {925, 1632}, {927, 4573}, {929, 4560}, {930, 13290}, {931, 4612}, {933, 13318}, {934, 1414}, {942, 1175}, {946, 37369}, {952, 3109}, {960, 1798}, {972, 1817}, {974, 1181}, {1010, 24809}, {1014, 15728}, {1019, 1308}, {1043, 2370}, {1055, 5060}, {1062, 9638}, {1098, 2975}, {1101, 2605}, {1109, 2606}, {1113, 2574}, {1114, 2575}, {1125, 13605}, {1151, 7601}, {1152, 7602}, {1154, 2070}, {1155, 11670}, {1157, 15907}, {1169, 2300}, {1171, 2054}, {1172, 8759}, {1173, 1493}, {1193, 38453}, {1194, 34945}, {1199, 5462}, {1205, 11574}, {1209, 9379}, {1216, 7512}, {1289, 2409}, {1291, 1510}, {1292, 2775}, {1293, 2776}, {1295, 2778}, {1296, 2434}, {1301, 2445}, {1302, 2407}, {1309, 5379}, {1310, 4592}, {1316, 2782}, {1320, 31523}, {1329, 9701}, {1340, 6142}, {1341, 6141}, {1370, 1660}, {1379, 5638}, {1380, 5639}, {1384, 9486}, {1396, 2376}, {1397, 17127}, {1408, 8686}, {1412, 1477}, {1415, 32693}, {1428, 7292}, {1434, 2369}, {1469, 32289}, {1473, 26910}, {1474, 9085}, {1478, 18968}, {1479, 12896}, {1498, 2935}, {1499, 2696}, {1506, 9697}, {1509, 2368}, {1531, 10296}, {1550, 36170}, {1553, 36172}, {1568, 3153}, {1593, 11439}, {1594, 9820}, {1599, 10132}, {1600, 10133}, {1619, 2063}, {1621, 2185}, {1633, 13397}, {1641, 10717}, {1649, 20404}, {1656, 9704}, {1657, 34584}, {1658, 18436}, {1691, 3231}, {1692, 3291}, {1698, 9587}, {1699, 9586}, {1793, 6326}, {1812, 26703}, {1853, 30744}, {1870, 19469}, {1897, 7452}, {1936, 23692}, {1946, 2714}, {1963, 19133}, {1971, 3289}, {1977, 18268}, {1992, 9084}, {2004, 21461}, {2005, 21462}, {2030, 9136}, {2051, 9562}, {2071, 2693}, {2072, 25739}, {2076, 8627}, {2106, 9455}, {2112, 2311}, {2175, 6015}, {2176, 35105}, {2177, 28559}, {2187, 29056}, {2193, 32726}, {2203, 2991}, {2223, 2711}, {2330, 5297}, {2341, 16554}, {2353, 28710}, {2365, 6514}, {2366, 3926}, {2382, 5009}, {2384, 3285}, {2393, 11416}, {2482, 11006}, {2644, 37135}, {2651, 3218}, {2682, 36174}, {2691, 3309}, {2692, 3667}, {2694, 6001}, {2700, 17209}, {2704, 8642}, {2705, 8643}, {2708, 37793}, {2709, 8644}, {2710, 18860}, {2717, 5536}, {2719, 23224}, {2720, 23189}, {2721, 16702}, {2723, 14956}, {2724, 14953}, {2725, 18206}, {2726, 5211}, {2727, 4091}, {2729, 6629}, {2740, 4786}, {2741, 26006}, {2746, 30234}, {2748, 6065}, {2754, 25083}, {2760, 3977}, {2769, 6332}, {2790, 36192}, {2794, 36163}, {2858, 14588}, {2859, 15413}, {2860, 7199}, {2861, 17139}, {2862, 30941}, {2863, 30939}, {2867, 3265}, {2868, 3266}, {2883, 11744}, {2886, 9702}, {2888, 10274}, {2892, 5596}, {2909, 7763}, {2915, 22136}, {2916, 35218}, {2917, 32338}, {2929, 22534}, {2937, 6101}, {2966, 6037}, {3016, 6787}, {3049, 9091}, {3052, 28531}, {3053, 15504}, {3056, 32290}, {3068, 8998}, {3069, 13990}, {3098, 7492}, {3100, 10118}, {3101, 10119}, {3108, 11205}, {3111, 9153}, {3129, 5611}, {3130, 5615}, {3133, 8883}, {3146, 13202}, {3147, 11411}, {3154, 31945}, {3219, 3955}, {3220, 22128}, {3229, 14602}, {3230, 5006}, {3258, 17511}, {3288, 14509}, {3357, 25564}, {3398, 37338}, {3428, 22583}, {3431, 4550}, {3511, 23180}, {3515, 12164}, {3517, 12160}, {3519, 18282}, {3520, 12038}, {3522, 10990}, {3523, 10984}, {3524, 11693}, {3525, 13336}, {3526, 20379}, {3529, 25712}, {3532, 15748}, {3533, 38725}, {3541, 15115}, {3542, 6193}, {3548, 11457}, {3565, 16680}, {3567, 7506}, {3574, 18428}, {3576, 11709}, {3579, 5951}, {3581, 7575}, {3589, 25328}, {3616, 11735}, {3618, 15118}, {3619, 5157}, {3620, 19126}, {3627, 37495}, {3628, 13353}, {3699, 9059}, {3732, 4241}, {3746, 7343}, {3747, 12031}, {3763, 6698}, {3796, 5621}, {3815, 9604}, {3818, 5169}, {3819, 15246}, {3832, 11424}, {3845, 13482}, {3851, 15046}, {3868, 13739}, {3869, 14529}, {3877, 17512}, {3909, 4585}, {3917, 6030}, {3939, 4627}, {4074, 10328}, {4215, 29009}, {4235, 30247}, {4238, 26706}, {4239, 15988}, {4242, 30250}, {4273, 28317}, {4296, 19505}, {4414, 5197}, {4557, 4629}, {4559, 8687}, {4584, 5384}, {4590, 9150}, {4600, 8709}, {4602, 9065}, {4603, 30670}, {4625, 34083}, {5008, 31609}, {5020, 5422}, {5026, 5108}, {5033, 8617}, {5050, 5544}, {5055, 15088}, {5070, 20396}, {5085, 5646}, {5091, 24617}, {5092, 5650}, {5094, 7703}, {5102, 31860}, {5104, 9831}, {5107, 13192}, {5111, 20977}, {5133, 23292}, {5135, 37633}, {5137, 33129}, {5138, 14996}, {5170, 8649}, {5189, 29012}, {5235, 9093}, {5250, 37032}, {5254, 9603}, {5285, 38822}, {5320, 37685}, {5333, 9103}, {5398, 19245}, {5418, 9677}, {5446, 34484}, {5449, 14940}, {5463, 11612}, {5464, 11613}, {5477, 6388}, {5505, 8542}, {5548, 32686}, {5562, 7488}, {5587, 9621}, {5597, 13208}, {5598, 13209}, {5603, 11723}, {5606, 17404}, {5612, 6105}, {5616, 6104}, {5618, 9200}, {5619, 9201}, {5641, 9141}, {5649, 35909}, {5661, 38352}, {5840, 10767}, {5846, 32298}, {5886, 12261}, {5890, 6644}, {5891, 18475}, {5899, 13391}, {5907, 13367}, {5943, 12834}, {5946, 13358}, {5965, 32223}, {5994, 6137}, {5995, 6138}, {6003, 34961}, {6011, 9810}, {6032, 11187}, {6043, 32911}, {6070, 22104}, {6093, 14653}, {6102, 11561}, {6150, 15770}, {6153, 11536}, {6197, 12661}, {6198, 12888}, {6200, 35299}, {6214, 12804}, {6215, 12803}, {6225, 30552}, {6233, 13242}, {6236, 32228}, {6240, 22660}, {6243, 37440}, {6284, 12374}, {6353, 6515}, {6368, 30716}, {6396, 35300}, {6413, 10962}, {6414, 10960}, {6467, 34470}, {6516, 13395}, {6528, 18831}, {6561, 9676}, {6564, 35834}, {6565, 35835}, {6574, 7259}, {6583, 18180}, {6617, 15324}, {6638, 23606}, {6640, 23294}, {6642, 7592}, {6676, 37636}, {6677, 11245}, {6739, 36154}, {6777, 30465}, {6778, 30468}, {6795, 9159}, {6816, 18925}, {6997, 11427}, {7058, 17135}, {7085, 26911}, {7256, 8706}, {7341, 36277}, {7354, 12373}, {7391, 31383}, {7417, 10753}, {7465, 37659}, {7476, 10101}, {7482, 10098}, {7487, 15473}, {7502, 23039}, {7503, 15056}, {7517, 16266}, {7519, 31670}, {7526, 15058}, {7527, 11430}, {7533, 18427}, {7542, 31831}, {7550, 10170}, {7556, 37478}, {7574, 19381}, {7577, 18474}, {7583, 19052}, {7584, 19051}, {7605, 25555}, {7666, 33541}, {7669, 14060}, {7689, 21844}, {7708, 8585}, {7715, 11566}, {7724, 10902}, {7725, 11824}, {7726, 11825}, {7779, 19558}, {7796, 18796}, {7799, 35568}, {8024, 19571}, {8025, 9108}, {8029, 12064}, {8041, 10329}, {8057, 22239}, {8151, 14695}, {8154, 14889}, {8200, 12466}, {8207, 12467}, {8220, 12894}, {8221, 12895}, {8227, 9622}, {8286, 24916}, {8287, 24955}, {8371, 30220}, {8537, 12596}, {8550, 37648}, {8584, 20192}, {8651, 10425}, {8673, 10423}, {8675, 9060}, {8681, 32127}, {8683, 8698}, {8690, 23831}, {8691, 23889}, {8705, 10510}, {8722, 34095}, {8758, 18609}, {8787, 9169}, {8911, 26912}, {8939, 19412}, {8943, 19413}, {8976, 13915}, {8994, 9540}, {9019, 19596}, {9027, 10102}, {9044, 9186}, {9064, 23347}, {9066, 23342}, {9073, 33295}, {9075, 30966}, {9080, 9182}, {9082, 27644}, {9101, 16285}, {9110, 18166}, {9126, 19902}, {9130, 9184}, {9160, 14270}, {9161, 23217}, {9162, 9202}, {9163, 9203}, {9171, 34539}, {9177, 15566}, {9208, 11636}, {9214, 10555}, {9301, 37914}, {9308, 37070}, {9514, 11794}, {9591, 31737}, {9626, 31738}, {9653, 10895}, {9666, 10896}, {9730, 11806}, {9781, 13861}, {9811, 13486}, {9818, 38396}, {9830, 14833}, {9833, 37444}, {9862, 35922}, {9919, 11414}, {9927, 16868}, {9937, 12271}, {9973, 15140}, {9996, 12501}, {10018, 12359}, {10111, 18912}, {10112, 14049}, {10203, 10227}, {10225, 15767}, {10263, 18378}, {10276, 34989}, {10294, 36852}, {10295, 15136}, {10298, 11202}, {10301, 21850}, {10303, 37515}, {10310, 12327}, {10516, 32274}, {10519, 32247}, {10533, 11417}, {10534, 11418}, {10537, 20243}, {10594, 11387}, {10601, 17809}, {10605, 15078}, {10610, 14128}, {10625, 12088}, {10627, 13564}, {10632, 10662}, {10633, 10661}, {10639, 32586}, {10640, 32585}, {10665, 10881}, {10666, 10880}, {10681, 11420}, {10682, 11421}, {10754, 25047}, {10766, 35901}, {10796, 12201}, {10942, 12905}, {10943, 12906}, {10989, 11645}, {11050, 15354}, {11078, 32461}, {11092, 32460}, {11107, 12528}, {11145, 13349}, {11146, 13350}, {11159, 11162}, {11176, 36255}, {11186, 13170}, {11199, 32437}, {11204, 35493}, {11248, 12381}, {11249, 12382}, {11250, 18439}, {11270, 33556}, {11328, 11842}, {11381, 12086}, {11423, 15024}, {11426, 15465}, {11499, 12334}, {11550, 31074}, {11559, 18364}, {11585, 12419}, {11656, 12243}, {11704, 34114}, {11793, 37126}, {11805, 15800}, {11820, 21312}, {11822, 12365}, {11823, 12366}, {11826, 12371}, {11827, 12372}, {11828, 12377}, {11829, 12378}, {11849, 35195}, {11898, 19154}, {12039, 29959}, {12068, 12079}, {12082, 37483}, {12084, 12290}, {12100, 22250}, {12113, 23240}, {12163, 32534}, {12225, 32330}, {12274, 12978}, {12275, 12979}, {12289, 18404}, {12293, 18504}, {12380, 15091}, {12588, 32243}, {12589, 32297}, {13015, 13055}, {13016, 13056}, {13163, 22051}, {13323, 16865}, {13348, 16661}, {13363, 15037}, {13364, 15038}, {13504, 15959}, {13509, 14961}, {13518, 19120}, {13558, 14687}, {13567, 19142}, {13582, 34306}, {13588, 15323}, {13596, 16194}, {13637, 13643}, {13757, 13762}, {13863, 30210}, {13935, 13969}, {13951, 13979}, {14008, 17188}, {14061, 15359}, {14133, 33021}, {14153, 20965}, {14247, 32027}, {14357, 36833}, {14360, 14928}, {14499, 14807}, {14500, 14808}, {14508, 36164}, {14528, 33537}, {14538, 34008}, {14539, 34009}, {14586, 32692}, {14590, 14696}, {14605, 35297}, {14606, 38366}, {14616, 24346}, {14651, 33511}, {14652, 34218}, {14658, 35006}, {14733, 21789}, {14781, 23105}, {14913, 21637}, {14933, 16169}, {14966, 26714}, {14981, 18337}, {14987, 37227}, {14998, 23969}, {15026, 32136}, {15028, 36752}, {15047, 22462}, {15069, 37638}, {15082, 20190}, {15138, 19374}, {15311, 16386}, {15322, 35342}, {15345, 32749}, {15431, 19124}, {15438, 37201}, {15439, 23067}, {15453, 35189}, {15535, 38224}, {15545, 15561}, {15731, 35997}, {15766, 36254}, {16039, 20626}, {16105, 37498}, {16168, 36193}, {16176, 25331}, {16238, 26879}, {16261, 31861}, {16270, 19347}, {16659, 23335}, {16806, 35337}, {16807, 35336}, {16876, 28838}, {16981, 37517}, {17167, 26708}, {17187, 28485}, {17222, 33628}, {17414, 32694}, {17455, 34079}, {17524, 28173}, {17714, 37484}, {17821, 17835}, {17977, 32849}, {18331, 33512}, {18358, 37454}, {18381, 32743}, {18403, 30522}, {18435, 18570}, {18488, 35482}, {18829, 33514}, {18916, 18932}, {19118, 19588}, {19119, 32241}, {19123, 32249}, {19125, 32251}, {19129, 32272}, {19131, 32275}, {19132, 32276}, {19134, 32293}, {19135, 32294}, {19137, 32300}, {19167, 19189}, {19171, 32258}, {19185, 19195}, {19406, 19507}, {19407, 19508}, {19424, 19482}, {19425, 19483}, {19440, 19484}, {19441, 19485}, {19459, 26206}, {19478, 22758}, {19576, 36214}, {19924, 37901}, {20063, 29317}, {20301, 38317}, {20766, 20877}, {20987, 26284}, {20999, 36942}, {21008, 25435}, {21294, 23674}, {21444, 23216}, {21663, 37941}, {22135, 33652}, {22333, 22334}, {22528, 36982}, {22648, 33499}, {22649, 33501}, {23208, 28724}, {23235, 31854}, {23344, 28210}, {23390, 23861}, {23582, 32725}, {23698, 36181}, {23700, 35296}, {23832, 28218}, {23878, 36886}, {23964, 32687}, {23967, 36904}, {23997, 29055}, {24000, 36068}, {24041, 36066}, {24281, 30927}, {24542, 25536}, {24550, 29310}, {24987, 37152}, {25577, 28868}, {25738, 26917}, {26156, 26926}, {26227, 27958}, {26257, 35301}, {26733, 36075}, {26874, 26880}, {26875, 26886}, {26889, 27003}, {26890, 27065}, {26898, 37068}, {28624, 32656}, {28708, 36851}, {29181, 37900}, {29300, 37619}, {29330, 36015}, {29352, 35983}, {30212, 37964}, {30528, 32733}, {30715, 30717}, {31133, 36990}, {31378, 31379}, {31626, 32078}, {31874, 34209}, {32072, 32563}, {32073, 32570}, {32074, 32434}, {32124, 37969}, {32225, 37907}, {32262, 37485}, {32264, 34774}, {32267, 37909}, {32315, 32599}, {32428, 32439}, {32445, 32547}, {32515, 37906}, {32608, 37922}, {32640, 32681}, {32676, 32691}, {33543, 35446}, {33586, 37672}, {33852, 33883}, {33873, 33876}, {34146, 37929}, {34150, 36169}, {34380, 37897}, {34381, 37963}, {34382, 37777}, {34394, 36759}, {34395, 36760}, {34783, 37814}, {34830, 38535}, {35002, 37916}, {35056, 35057}, {35447, 38623}, {36034, 36064}, {36065, 36084}, {36070, 36142}, {36078, 36134}, {37496, 37924}, {37649, 37990}

▸ Functional Relations94 relations

• X₁₁₀ = midpoint of X_i and X_j for these (i,j): (3, 399), (20, 146), (23, 323), (1495, 3292)

• X₁₁₀ = reflection of X_i in X_j for these (i,j): (3, 1511), (4, 113), (23, 1495), (67, 141), (74, 3), (265, 5), (382, 1539), (895, 6), (1177, 206), (3580, 468)

• X₁₁₀ = circumcircle-antipode of X₇₄

• X₁₁₀ = isogonal conjugate of X₅₂₃

• X₁₁₀ = isotomic conjugate of X₈₅₀

• X₁₁₀ = isogonal conjugate of the isotomic conjugate of X₉₉

• X₁₁₀ = inverse of X₂ in the Brocard circle

• X₁₁₀ = complement of X₃₄₄₈

• X₁₁₀ = anticomplement of X₁₂₅

• X₁₁₀ = X_i-Ceva conjugate of X_j for these (i,j): (249, 6), (250, 3)

• X₁₁₀ = cevapoint of X_i and X_j for these (i,j): (3, 520), (5, 523), (6, 512), (141, 525)

• X₁₁₀ = X_i-cross conjugate of X_j for these (i,j): (1, 59), (3, 250), (6, 249), (109, 162), (351, 111), (512, 6), (520, 3), (523, 54), (526, 74)

• X₁₁₀ = crosssum of X_i and X_j for these (i,j): (2, 148), (512, 647), (520, 647)

• X₁₁₀ = crossdifference of every pair of points on line X₁₁₅X₁₂₅

• X₁₁₀ = X_i-Hirst inverse of X_j for these (i,j): (1, 245), (2, 125), (3, 246), (4, 247)

• X₁₁₀ = X_i-beth conjugate of X_j for these (i,j): (21, 759), (643, 643)

• X₁₁₀ = X₂₃-of-1st-Brocard triangle

• X₁₁₀ = X₁₁₁-of-circumsymmedial-triangle

• X₁₁₀ = X₃₂₃-of-orthocentroidal-triangle

• X₁₁₀ = X₁₃₇-of-excentral-triangle

• X₁₁₀ = X₁₂₈-of-hexyl-triangle

• X₁₁₀ = trilinear pole of the Brocard axis

• X₁₁₀ = trilinear pole of PU(29) (see ETC→Tables→Bicentric Pairs)

• X₁₁₀ = crosssum of polar circle intercepts of Euler line

• X₁₁₀ = perspector of ABC and vertex-triangle of anticevian triangles of X₃ and X₆

• X₁₁₀ = Johnson-circumconic antipode of X₂₆₅

• X₁₁₀ = MacBeath-circumconic antipode of X₈₉₅

• X₁₁₀ = perspector of conic {A, B, C,PU(2)}

• X₁₁₀ = intersection of trilinear polars of P(2) and U(2)

• X₁₁₀ = intersection of tangents to Steiner circumellipse at X₉₉ and X₆₄₈

• X₁₁₀ = crosspoint of X₉₉ and X₆₄₈

• X₁₁₀ = reflection of X₄₇₆ in the Euler line

• X₁₁₀ = reflection of X₆₉₁ in the Brocard axis

• X₁₁₀ = reflection of X₂₃ in the Lemoine axis

• X₁₁₀ = reflection of X₁₂₉₀ in line X₁X₃

• X₁₁₀ = reflection of X₁₁₁ in line X₃X₃₅₁

• X₁₁₀ = inverse-in-polar-circle of X₁₃₆

• X₁₁₀ = inverse-in-{circumcircle, nine-point circle}-inverter of X₁₁₄

• X₁₁₀ = inverse-in-Moses-radical-circle of X₂₇₁₅

• X₁₁₀ = inverse-in-O(15,16) of X₈₄₃, where O(15,16) is the circle having segment X₁₅X₁₆ as diameter

• X₁₁₀ = X_i-isoconjugate of X_j for these (i,j): (6, 1577), (92, 647), (1577, 6)

• X₁₁₀ = perspector of circumorthic triangle and Johnson triangle (the reflection triangles of X₄ and X₃, resp.)

• X₁₁₀ = trilinear product of vertices of circumtangential triangle

• X₁₁₀ = {X₃,X₁₅₆}-harmonic conjugate of X₁₆₁₄

• X₁₁₀ = orthocentroidal-to-ABC similarity image of X₂

• X₁₁₀ = 4th-Brocard-to-circumsymmedial similarity image of X₂

• X₁₁₀ = isogonal conjugate of isotomic conjugate of trilinear pole of line X₂X₆

• X₁₁₀ = the point of intersection, other than A, B, C, of the circumcircle and Johnson circumconic

• X₁₁₀ = the point of intersection, other than A, B, C, of the circumcircle and MacBeath circumconic

• X₁₁₀ = the point of intersection, other than A, B, C, of the circumcircle and circumconic {A, B, C,PU(5)}}

• X₁₁₀ = Collings transform of X₅

• X₁₁₀ = Collings transform of X₆

• X₁₁₀ = intersection of tangents at X₆₁ and X₆₂ to the Napoleon-Feuerbach cubic K005

• X₁₁₀ = SR(PU(4))

• X₁₁₀ = insimilicenter of nine-point circle and sine-triple-angle circle

• X₁₁₀ = insimilicenter of circumcircle and nine-point circle of tangential triangle; the exsimilicenter is X₁₆₁₄

• X₁₁₀ = X₇₉₇₂-of-Trinh-triangle

• X₁₁₀ = Ψ(X_i, X_j) for these (i,j): (1, 21), (2, 6), (3, 49), (4, 2), (5, 51), (6, 3), (7, 21), (8, 21), (9, 21), (19, 1), (53, 5), (54, 3), (64, 3), (65, 1), (66, 3), (67, 3), (68, 3), (69, 3), (73, 3), (74, 3), (75, 1), (76, 2), (93, 4), (95, 54), (115, 125), (190, 99), (250, 186)

• X₁₁₀ = X₁₁₀-of-1st-Parry-triangle

• X₁₁₀ = X₇₄-of-2nd-Parry-triangle

• X₁₁₀ = center of similitude of ABC and 1st Parry triangle

• X₁₁₀ = inverse-in-Parry-isodynamic-circle of X₁₁₁; see X₂

• X₁₁₀ = barycentric product of PU(i) for these i: 78, 145

• X₁₁₀ = perspector of unary cofactor triangles of outer and inner Napoleon triangles

• X₁₁₀ = X₆₇₉₂-of-4th-anti-Brocard-triangle

• X₁₁₀ = X₁₁₁-of-anti-Artzt-triangle

• X₁₁₀ = perspector of circumcevian triangle of X₃₆ and cross-triangle of ABC and 2nd circumperp triangle

• X₁₁₀ = perspector of circumcevian triangle of X₁₈₇ and cross-triangle of ABC and circumsymmedial triangle

• X₁₁₀ = Kiepert image of X₃

• X₁₁₀ = Jerabek image of X₂

• X₁₁₀ = Cundy-Parry Phi transform of X₁₄₂₆₄

• X₁₁₀ = endo-homothetic center of X₂-altimedial and X₂-anti-altimedial triangles

• X₁₁₀ = endo-homothetic center of X₂₀-altimedial and X₃-anti-altimedial triangles

• X₁₁₀ = Thomson isogonal conjugate of X₃₀

• X₁₁₀ = Lucas isogonal conjugate of X₃₀

• X₁₁₀ = center of the perspeconic of these triangles: inner and outer Napoleon

• X₁₁₀ = intersection of antipedal lines of X₁₁₁₃ and X₁₁₁₄

• X₁₁₀ = X₁₀₄-of-circumorthic-triangle if ABC is acute

• X₁₁₀ = perspector, wrt circumorthic triangle, of polar circle

• X₁₁₀ = trilinear product of circumcircle intercepts of line X₁X₂₁

• X₁₁₀ = barycentric product of circumcircle intercepts of line X₂X₆

• X₁₁₀ = barycentric product X₆*X₉₉

• X₁₁₀ = barycentric product of Steiner circumellipse intercepts of Brocard axis

• X₁₁₀ = Feuerbach point of the tangential triangle if ABC is acute; otherwise, a vertex of the Feuerbach triangle of the tangential triangle

• X₁₁₀ = barycentric product of MacBeath circumconic intercepts of Euler line

• X₁₁₀ = excentral-to-ABC functional image of X₁₇₆₈

• X₁₁₀ = intouch-to-ABC functional image of X₁₁

• X₁₁₀ = trilinear pole, wrt circumtangential triangle, of Euler line

• X₁₁₀ = trilinear pole, wrt 1st Parry triangle, of Lemoine axis

• X₁₁₀ = trilinear product of circumcircle intercepts of line X₁X₂₁

• X₁₁₀ = de-Longchamps-circle-inverse of X₃₄₁₈₆

• X₁₁₀ = antipode of X₃₅₈₀ in Walsmith rectangular hyperbola

• X₁₁₀ = orthocenter of X₆X₇₄X₃₅₆₉

• X₁₁₀ = Conway-circle-inverse of X₃₈₄₈₀

Notes

X₁₁₀ is the center of the Stammler hyperbola, SH, which is the rectangular hyperbola that passes through X₁, X₃, X₆, X₁₅₅, X₁₅₉, X₁₉₅, X₃₉₉, X₁₄₉₈, X₂₉₁₆, X₂₉₁₇, X₂₉₁₈, X₂₉₂₉, X₂₉₃₀, X₂₉₃₁, X₂₉₃₅, X₂₉₄₈, X₃₅₁₁, the excenters, and the vertices of the tangential triangle. SH is the bianticevian conic of X₁ and X₆ and the antipedal-anticevian conic of X₃. SH is the tangential isogonal conjugate of the Euler line, the tangential isotomic conjugate of the van Aubel line, the excentral isogonal conjugate of line X₃₀X₄₀, and the excentral isotomic conjugate of line X₁₉₁X₂₉₃₈. SH is tangent to the Euler line at X₃ and meets the line at infinity (and the Jerabek hyperbola, other than at X₃ and X₆) at X₂₅₇₄ and X₂₅₇₅. SH is the locus of a point P for which the P-Brocard triangle is perspective to ABC. (Randy Hutson, 9/23/2011, 1/29/2015)

J. W. Clawson, “Points on the circumcircle,” American Mathematical Monthly 32 (1925) 169-174.

Roland H. Eddy and R. Fritsch, “The conics of Ludwig Kiepert: a comprehensive lesson in the geometry of the triangle,” Mathematics Magazine 67 (1994) 188-205.

Benedetto Scimemi, “Paper-folding and Euler’s Theorem Revisited,” :cite:`Forum Geometricorum <forum-geometricorum>`.

Seven constructions from Randy Hutson, January 29, 2015:

1. Let P be a point, other than X₄, on Euler line. Let A' be the reflection of P in BC, and define B' and C' cyclically. The circumcircles of AA'B', BC'A', and CA'B' concur in X₁₁₀.

2. Let Q be a point on line X₂X₆ other than X₂. Let A'B'C' be the cevian triangle of Q. Let A″ be the {B,C}-harmonic conjugate of A' (or equivalently, A″ = BC∩B'C'), and define B″, C″ cyclically. The circumcircles of AB″C″, BC″A″, CA″B″ concur in X₁₁₀.

3. Let L_a, L_b, L_c be the lines through A, B, C, resp. parallel to the Euler line. Let M_a, M_b, M_c be the reflections of BC, CA, AB in L_a, L_b, L_c, resp. Let A' = M_b∩M_c, and define B', C' cyclically. Triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. Let A″B″C″ be the reflection of A'B'C' in the Euler line. The triangle A″B″C″ is homothetic to ABC, with center of homothety X₁₂₅ and centroid X₁₁₀. (See Hyacinthos #16741/16782, Sep 2008.)

4. Let A', B', C' be the intersections of the Lemoine axis and lines BC, CA, AB, resp. The circumcircles of AB'C', BC'A', CA'B' concur in X₁₁₀.

5. Let A', B', C' be the intersections of line X₃₆X₂₃₈ and lines BC, CA, AB, resp. The circumcircles of AB'C', BC'A', CA'B' concur in X₁₁₀.

6. Let A', B', C' be the intersections of the Euler line and lines BC, CA, AB, resp. Let O_a, O_b, O_c be the circumcenters of AB'C', BC'A', CA'B', resp. The lines AO_a, BO_b, CO_c concur in X₁₁₀.

7. Let N_a be the reflection of X₅ in the perpendicular bisector of BC, and define N_b, N_c cyclically. X₁₁₀ = X₂₀₇₀ of N_aN_bN_c.

Let A₂B₂C₂ and A₃B₃C₃ be the 2nd and 3rd Parry triangles. Let A' be the barycentric product A₂*A₃, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₁₁₀. (Randy Hutson, February 10, 2016)

Let A'B'C' be the triangle described in ADGEOM #2697 (8/26/2015, Tran Quang Hung). A'B'C' is homothetic to the anticomplementary triangle at X₁₁₀. The nine-point circle of A'B'C' touches the circumcircle of ABC at X₁₁₀. Also, X₁₁₀ = X₁₂₅-of-A'B'C'. (Randy Hutson, Novermber 2, 2016)

Let A' be the trilinear pole of the perpendicular bisector of BC, and define B', C' cyclically. A'B'C' is homothetic to the circumanticevian triangle of X₄ at X₁₁₀. X₁₁₀ is also the barycentric product A'*B'*C'. (Randy Hutson, January 29, 2018)

Let O_A be the circle centered at the A-vertex of the AAOA triangle and passing through A; define O_B and O_C cyclically. Circles O_A, O_B, O_C concur in X₁₁₀. (Randy Hutson, August 30, 2020)

X₁₁₀ is the {X₅,X₄₉}-harmonic conjugate of X₅₄.