X₁₀₅

Λ(INCENTER, SYMMEDIAN POINT)

X₁₀₅ = anticomplement of X₁₂₀

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

X₁₀₅ (Λ(INCENTER, SYMMEDIAN POINT)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 11}, {1, 41}

• Related Centers: X₅₁₈ (ISOGONAL CONJUGATE OF X₁₀₅), X₂ (CENTROID), X₁₁ (FEUERBACH POINT), X₁ (INCENTER), X₄₁ (-CEVA CONJUGATE OF X₃₁)

Trilinears: \(1/[b^{2} + c^{2} - a(b + c)] : 1/[c^{2} + a^{2} - b(c + a)] : 1/[a^{2} + b^{2} - c(a + b)]\)

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

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

Search Value (6, 9, 13): \(2.693929186947\)

▸ Incident Central Lines291 lines

{1, 41}, {2, 11}, {3, 277}, {4, 5511}, {5, 10743}, {6, 1002}, {7, 3423}, {8, 16048}, {9, 4712}, {10, 20656}, {20, 34547}, {21, 99}, {22, 13397}, {23, 1290}, {25, 108}, {28, 112}, {30, 2691}, {31, 57}, {36, 1308}, {37, 1390}, {44, 6017}, {56, 279}, {65, 1170}, {74, 2775}, {75, 20628}, {81, 110}, {88, 901}, {89, 4588}, {98, 2788}, {102, 2814}, {103, 2254}, {104, 885}, {106, 1022}, {107, 2833}, {111, 2837}, {115, 37014}, {165, 1054}, {175, 30385}, {176, 30386}, {183, 9067}, {186, 10101}, {230, 9090}, {238, 291}, {243, 1309}, {330, 932}, {335, 6652}, {385, 33674}, {386, 3034}, {392, 29127}, {404, 6012}, {405, 26035}, {468, 2766}, {474, 34124}, {476, 7469}, {511, 2704}, {512, 2711}, {513, 840}, {514, 2725}, {515, 2730}, {516, 2736}, {517, 2742}, {518, 1280}, {519, 2748}, {520, 2749}, {521, 2750}, {522, 2751}, {523, 2752}, {524, 2753}, {525, 2754}, {551, 8691}, {595, 6577}, {612, 3722}, {644, 1083}, {651, 1362}, {659, 676}, {663, 12032}, {666, 898}, {667, 14665}, {691, 1325}, {693, 2862}, {739, 16501}, {743, 29956}, {789, 18031}, {805, 2106}, {825, 985}, {831, 1125}, {835, 37325}, {904, 1201}, {905, 28838}, {907, 17560}, {910, 919}, {927, 1447}, {931, 37870}, {935, 2074}, {957, 16483}, {958, 1219}, {959, 1191}, {960, 1257}, {961, 1104}, {984, 28883}, {993, 29351}, {995, 29067}, {999, 14074}, {1024, 1635}, {1036, 1310}, {1064, 36516}, {1086, 1633}, {1123, 6135}, {1156, 14191}, {1193, 27667}, {1200, 9445}, {1224, 5259}, {1255, 3748}, {1261, 7081}, {1283, 4220}, {1284, 8852}, {1289, 30733}, {1294, 9520}, {1295, 9521}, {1296, 4221}, {1297, 9523}, {1304, 37963}, {1305, 2218}, {1319, 14733}, {1320, 5376}, {1332, 25048}, {1336, 6136}, {1422, 2208}, {1428, 9455}, {1456, 24016}, {1458, 23694}, {1477, 37626}, {1486, 4000}, {1724, 29303}, {1929, 2702}, {1946, 2724}, {1995, 9058}, {2006, 2078}, {2108, 15485}, {2113, 9500}, {2144, 20332}, {2191, 7289}, {2223, 6185}, {2224, 30117}, {2264, 5572}, {2282, 28624}, {2292, 29119}, {2306, 36072}, {2347, 9440}, {2402, 9061}, {2651, 6083}, {2696, 37960}, {2703, 17946}, {2717, 37815}, {2720, 3660}, {2737, 6909}, {2743, 5121}, {2864, 16087}, {2865, 3262}, {2866, 35519}, {2982, 5173}, {2990, 6099}, {3056, 37659}, {3263, 3685}, {3309, 28914}, {3322, 7336}, {3428, 15747}, {3485, 13395}, {3565, 16049}, {3576, 15746}, {3601, 6575}, {3651, 30257}, {3659, 7589}, {3757, 8707}, {3827, 35185}, {3871, 32019}, {3924, 7132}, {4227, 30247}, {4231, 30250}, {4232, 9107}, {4236, 24617}, {4239, 9070}, {4310, 24320}, {4458, 26702}, {4511, 29241}, {4578, 24820}, {4585, 10755}, {4859, 24309}, {4904, 18343}, {5020, 15252}, {5089, 36122}, {5091, 35280}, {5205, 6079}, {5211, 25495}, {5222, 37580}, {5248, 6013}, {5262, 29143}, {5268, 28226}, {5273, 25494}, {5297, 28210}, {5310, 37261}, {5322, 6186}, {5323, 5545}, {5338, 32691}, {5695, 31130}, {5992, 31129}, {6014, 35445}, {6016, 20331}, {6081, 14203}, {6169, 11051}, {6180, 9309}, {6244, 16434}, {6553, 12513}, {7077, 23612}, {7175, 20978}, {7179, 8543}, {7465, 34879}, {7766, 29227}, {7965, 37456}, {8297, 8298}, {8647, 9441}, {8652, 17024}, {8684, 33676}, {8697, 26745}, {8699, 36603}, {8708, 16830}, {8709, 8851}, {8848, 16363}, {9057, 26246}, {9059, 26227}, {9068, 26281}, {9072, 9465}, {9073, 26277}, {9081, 26249}, {9086, 26229}, {9089, 26239}, {9096, 26279}, {9104, 26245}, {9317, 28850}, {9343, 9344}, {9456, 16507}, {9508, 28471}, {9511, 15731}, {10098, 37961}, {10246, 28536}, {10387, 25878}, {10789, 30554}, {10980, 28162}, {11284, 23858}, {12329, 37650}, {12589, 26540}, {13245, 29310}, {13444, 14596}, {13595, 26711}, {14625, 19309}, {15382, 34381}, {15571, 20045}, {16466, 36080}, {16485, 26715}, {16487, 26716}, {16684, 37033}, {16693, 20470}, {16752, 16876}, {20219, 24929}, {21214, 28469}, {21511, 23407}, {21793, 28899}, {23536, 28029}, {23834, 28583}, {24174, 28575}, {24178, 37328}, {24325, 24339}, {24540, 24669}, {24808, 38665}, {25304, 26657}, {26232, 26238}, {26237, 26243}, {27789, 28196}, {28156, 30350}, {28477, 37399}, {28480, 37431}, {28879, 38316}, {28895, 38315}, {29048, 30148}, {29363, 32911}, {29681, 33637}, {32630, 37587}, {33654, 36073}, {34084, 34085}, {34446, 37541}, {36740, 38053}

▸ Functional Relations27 relations

• X₁₀₅ = reflection of X_i in X_j for these (i,j): (644, 1083), (1292, 3)

• X₁₀₅ = isogonal conjugate of X₅₁₈

• X₁₀₅ = anticomplement of X₁₂₀

• X₁₀₅ = cevapoint of X₁ and X₂₃₈

• X₁₀₅ = X₁-Hirst inverse of X₂₉₄

• X₁₀₅ = X_i-beth conjugate of X_j for these (i,j): (21, 101), (927, 105)

• X₁₀₅ = Λ(X₁, X₆)

• X₁₀₅ = isotomic conjugate of X₃₂₆₃

• X₁₀₅ = crossdifference of every pair of points on line X₆₆₅X₁₆₄₂

• X₁₀₅ = Ψ(X_i, X_j) for these (i,j): (1, 514), (2, 650), (6, 513), (57, 649), (59, 651), (100, 2), (101, 1), (190, 9), (650, 11), (651, 6)

• X₁₀₅ = reflection of X₂₇₅₂ in the Euler line

• X₁₀₅ = reflection of X₂₇₁₁ in the Brocard axis

• X₁₀₅ = reflection of X₈₄₀ in line X₁X₃

• X₁₀₅ = X₁₃₂-of-excentral-triangle

• X₁₀₅ = X₁₂₇-of-hexyl-triangle

• X₁₀₅ = X₆-isoconjugate of X₃₉₁₂

• X₁₀₅ = inverse-in-{circumcircle, nine-point circle}-inverter of X₁₁

• X₁₀₅ = trilinear pole of PU(i) for these i: 46, 54

• X₁₀₅ = trilinear product of PU(96)

• X₁₀₅ = bicentric sum of PU(142)

• X₁₀₅ = Thomson-isogonal conjugate of X₃₃₀₉

• X₁₀₅ = Lucas-isogonal conjugate of X₃₃₀₉

• X₁₀₅ = polar conjugate of isogonal conjugate of X₃₂₆₅₈

• X₁₀₅ = trilinear product of circumcircle intercepts of line X₁X₅₁₄

• X₁₀₅ = trilinear pole, wrt Thomson triangle, of line X₃₅₄X₆₁₂

• X₁₀₅ = Conway-circle-inverse of X₃₈₄₇₉

• X₁₀₅ = X₁₉-isoconjugate of X₂₅₀₈₃

Notes

X₁₀₅ is the perspector of ABC and the (degenerate) side-triangle of the circumcevian triangles of X₃₅₁₃ and X₃₅₁₄. (Randy Hutson, June 7, 2019)