X₁₀₁

Ψ(INCENTER, SYMMEDIAN POINT)

X₁₀₁ = anticomplement of X₁₁₆

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

X₁₀₁ (Ψ(INCENTER, SYMMEDIAN POINT)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {20, 152}

• Related Centers: X₂₀ (DE LONGCHAMPS POINT), X₁₅₂ (REFLECTION OF X₂₀ IN X₁₀₁)

Trilinears: \(a/(b - c) : b/(c - a) : c/(a - b)\)

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

Tripolars: \(|b c (b - c)| : :\)

Search Value (6, 9, 13): \(1.320506736352\)

▸ Alternate Coordinates2 forms

• Trilinears: \(a(a - b)(a - c) : b(b - c)(b - a) : c(c - a)(c - b)\)

• Barycentrics: \(a^{2} (\sin^{2} A + \sin B \sin C - \sin C \sin A - \sin A \sin B) : :\)

▸ Linear Combinations1 combination

• X₁₀₁ = \((r ^{2} + 6rR + 8R^{2} + s^{2})\) X₁ - \(6R(r + 4R)\) X₂ - \(2(r^{2} + 4rR - s^{2})\) X₃ (Peter Moses, April 2, 2013)

▸ Incident Central Lines586 lines

{1, 41}, {2, 116}, {3, 103}, {4, 118}, {5, 10739}, {6, 106}, {7, 2369}, {8, 1311}, {9, 48}, {10, 98}, {11, 10770}, {19, 913}, {20, 152}, {21, 3294}, {25, 3190}, {27, 22000}, {28, 3191}, {29, 22018}, {30, 2688}, {31, 609}, {32, 595}, {33, 20624}, {34, 2376}, {35, 1334}, {36, 672}, {37, 284}, {39, 21008}, {40, 972}, {42, 111}, {43, 9082}, {44, 2718}, {45, 2278}, {55, 2291}, {56, 218}, {57, 15728}, {58, 172}, {59, 657}, {63, 36016}, {71, 74}, {72, 2360}, {75, 767}, {78, 205}, {80, 19628}, {86, 2368}, {99, 190}, {100, 644}, {102, 198}, {107, 1897}, {108, 1783}, {109, 654}, {110, 163}, {112, 4249}, {119, 6506}, {140, 38774}, {142, 18162}, {145, 9083}, {154, 1260}, {162, 36077}, {165, 6602}, {171, 3997}, {182, 23095}, {184, 3046}, {187, 2712}, {199, 3690}, {200, 2187}, {212, 32726}, {226, 36019}, {228, 2249}, {230, 17734}, {238, 9454}, {239, 9073}, {242, 29016}, {269, 2377}, {281, 32706}, {306, 2373}, {312, 30901}, {313, 2367}, {321, 30905}, {329, 36023}, {346, 2370}, {354, 1174}, {376, 38773}, {382, 38767}, {386, 3033}, {404, 16549}, {476, 4024}, {477, 36026}, {480, 2371}, {501, 21879}, {511, 2700}, {512, 2702}, {513, 1308}, {514, 664}, {515, 2723}, {516, 2724}, {517, 910}, {518, 2725}, {519, 2726}, {520, 2727}, {521, 2728}, {522, 929}, {523, 2690}, {524, 2729}, {548, 38766}, {550, 38765}, {551, 16503}, {560, 713}, {574, 28563}, {579, 2178}, {580, 29015}, {583, 21773}, {584, 16777}, {594, 2372}, {604, 1743}, {607, 11399}, {610, 1295}, {631, 6712}, {643, 931}, {649, 901}, {650, 2222}, {651, 934}, {652, 2149}, {653, 4605}, {656, 2722}, {660, 1492}, {661, 1290}, {663, 919}, {667, 813}, {668, 789}, {673, 17761}, {677, 36136}, {689, 1978}, {691, 4079}, {692, 926}, {693, 2860}, {697, 1501}, {699, 30634}, {715, 2205}, {717, 18900}, {729, 1918}, {731, 4279}, {733, 904}, {743, 869}, {745, 21814}, {747, 14620}, {753, 2276}, {755, 21035}, {758, 3509}, {761, 984}, {765, 898}, {798, 2703}, {822, 2719}, {825, 34069}, {827, 4628}, {831, 4553}, {833, 7239}, {835, 3952}, {839, 4033}, {859, 14964}, {872, 2375}, {896, 2721}, {899, 9081}, {902, 2384}, {908, 2861}, {912, 8558}, {922, 35107}, {925, 36145}, {930, 36148}, {932, 4579}, {935, 4064}, {944, 6554}, {947, 14872}, {952, 1146}, {953, 2183}, {956, 37658}, {958, 3041}, {976, 9078}, {991, 24320}, {1012, 5781}, {1016, 4595}, {1025, 6183}, {1043, 21070}, {1054, 23622}, {1083, 8299}, {1100, 5049}, {1111, 9317}, {1113, 2576}, {1114, 2577}, {1125, 9108}, {1126, 17962}, {1141, 7110}, {1149, 9097}, {1191, 30435}, {1193, 5280}, {1201, 5299}, {1212, 1385}, {1250, 36738}, {1259, 2365}, {1262, 4091}, {1292, 1633}, {1293, 2429}, {1294, 2822}, {1296, 2824}, {1297, 2825}, {1300, 1826}, {1302, 36149}, {1304, 36131}, {1305, 4552}, {1309, 3239}, {1310, 1332}, {1317, 4534}, {1319, 2348}, {1324, 2708}, {1326, 12031}, {1376, 24264}, {1381, 2590}, {1382, 2591}, {1384, 3052}, {1407, 23089}, {1412, 4641}, {1420, 16572}, {1429, 3008}, {1473, 7123}, {1474, 22021}, {1475, 5563}, {1499, 2740}, {1500, 18755}, {1503, 2741}, {1565, 5845}, {1577, 2864}, {1580, 2664}, {1604, 3197}, {1615, 6244}, {1618, 2742}, {1634, 23161}, {1657, 38768}, {1691, 21830}, {1698, 9103}, {1729, 3868}, {1731, 8609}, {1753, 2910}, {1755, 2699}, {1757, 19554}, {1759, 3869}, {1762, 16577}, {1780, 2198}, {1781, 2171}, {1785, 2202}, {1790, 3219}, {1818, 3220}, {1819, 22005}, {1914, 2251}, {1924, 9266}, {1939, 20691}, {1944, 29069}, {1951, 20752}, {1953, 16547}, {1958, 3729}, {1959, 2856}, {1973, 3811}, {1981, 18026}, {2077, 2272}, {2092, 38453}, {2140, 2141}, {2160, 21863}, {2173, 2687}, {2175, 34247}, {2177, 9331}, {2182, 2716}, {2195, 9319}, {2210, 3009}, {2220, 16685}, {2238, 5291}, {2241, 9351}, {2245, 17796}, {2256, 4254}, {2259, 2294}, {2266, 11529}, {2268, 3731}, {2273, 2277}, {2275, 28574}, {2287, 21061}, {2295, 5277}, {2304, 5248}, {2308, 8700}, {2312, 2747}, {2317, 28219}, {2318, 5285}, {2325, 2757}, {2327, 5279}, {2332, 6198}, {2333, 3563}, {2344, 36480}, {2347, 38452}, {2359, 21033}, {2374, 4028}, {2388, 23398}, {2398, 9057}, {2425, 8059}, {2426, 26716}, {2481, 24455}, {2503, 20982}, {2646, 16601}, {2689, 3700}, {2692, 14321}, {2695, 7359}, {2697, 21017}, {2701, 3709}, {2731, 4521}, {2736, 3309}, {2737, 3667}, {2738, 6000}, {2739, 6001}, {2743, 4394}, {2751, 3693}, {2752, 3930}, {2758, 3943}, {2766, 8611}, {2768, 3712}, {2770, 4062}, {2802, 4919}, {2859, 14208}, {2862, 3912}, {2863, 4358}, {2870, 22310}, {2908, 27396}, {2948, 16562}, {2975, 16552}, {3002, 17966}, {3053, 14974}, {3061, 30144}, {3083, 9098}, {3084, 9099}, {3090, 38775}, {3091, 20401}, {3119, 5531}, {3146, 38769}, {3185, 29068}, {3208, 8715}, {3241, 9095}, {3257, 4604}, {3261, 31624}, {3290, 30117}, {3295, 4258}, {3496, 3878}, {3501, 15323}, {3508, 9417}, {3522, 33521}, {3579, 21872}, {3616, 9105}, {3635, 9106}, {3661, 9075}, {3678, 15168}, {3679, 4390}, {3681, 32664}, {3688, 23868}, {3691, 5258}, {3699, 4103}, {3720, 9110}, {3734, 4713}, {3747, 18266}, {3754, 23621}, {3781, 28844}, {3832, 38770}, {3870, 9061}, {3903, 30670}, {3915, 7031}, {3920, 9077}, {3923, 24480}, {3970, 34772}, {3990, 26701}, {4006, 4420}, {4040, 9323}, {4041, 9090}, {4051, 22837}, {4053, 12030}, {4055, 26717}, {4093, 8625}, {4109, 36974}, {4120, 15343}, {4128, 23648}, {4153, 7270}, {4169, 6558}, {4209, 14377}, {4268, 16885}, {4289, 16672}, {4362, 9074}, {4427, 30727}, {4436, 6013}, {4513, 5687}, {4517, 37586}, {4530, 7972}, {4554, 34083}, {4556, 4629}, {4565, 4627}, {4567, 4584}, {4578, 6574}, {4585, 13396}, {4588, 5549}, {4600, 9150}, {4775, 28875}, {4794, 36086}, {4859, 7225}, {4872, 5074}, {4893, 14513}, {4904, 26007}, {4998, 31286}, {5010, 28535}, {5060, 5127}, {5088, 10025}, {5120, 37519}, {5189, 24055}, {5199, 28236}, {5206, 28551}, {5226, 34929}, {5228, 37272}, {5239, 36941}, {5240, 36940}, {5247, 35108}, {5249, 34934}, {5267, 29308}, {5275, 30116}, {5282, 5692}, {5293, 7281}, {5313, 28539}, {5315, 21764}, {5378, 38367}, {5514, 37725}, {5603, 5819}, {5660, 33573}, {5752, 29218}, {5840, 10772}, {5881, 23058}, {5949, 14219}, {5951, 16553}, {6014, 8658}, {6017, 8656}, {6037, 36132}, {6065, 6078}, {6079, 30720}, {6163, 20981}, {6205, 9352}, {6386, 9065}, {6510, 34371}, {6545, 38019}, {6586, 23990}, {6633, 9089}, {6645, 17499}, {6647, 35102}, {6733, 13444}, {7012, 36140}, {7077, 19561}, {7079, 17857}, {7084, 9439}, {7115, 10397}, {7117, 13006}, {7119, 21077}, {7122, 35105}, {7193, 9321}, {7287, 28899}, {7296, 28505}, {7297, 17444}, {7368, 10310}, {7391, 24054}, {7760, 34063}, {8012, 15931}, {8301, 14839}, {8624, 16514}, {8666, 21384}, {8676, 35182}, {8687, 32736}, {8694, 34074}, {9059, 17780}, {9067, 23891}, {9076, 15523}, {9086, 21272}, {9094, 17018}, {9104, 23705}, {9264, 21788}, {9306, 14827}, {9404, 34921}, {10246, 34522}, {10267, 32561}, {10454, 27410}, {10571, 22131}, {10638, 36737}, {10703, 38345}, {10799, 18759}, {11349, 20367}, {11720, 17468}, {12437, 21096}, {13384, 34930}, {13597, 21012}, {14422, 32630}, {14439, 15015}, {15378, 35184}, {15586, 21864}, {15654, 20471}, {15792, 21816}, {15817, 15830}, {16283, 20995}, {16560, 16578}, {16680, 28847}, {16787, 30148}, {16822, 33945}, {16946, 21769}, {16970, 37817}, {16997, 30114}, {17062, 31284}, {17137, 29473}, {17170, 26658}, {17205, 18723}, {17277, 18042}, {17284, 25940}, {17349, 27348}, {17454, 19302}, {17743, 27091}, {17744, 33299}, {17793, 24294}, {17798, 20683}, {20269, 30617}, {20336, 30920}, {20696, 21791}, {20780, 28914}, {20834, 21795}, {20999, 23988}, {21049, 37730}, {21138, 24281}, {21208, 26273}, {21232, 24685}, {21253, 23674}, {21285, 28734}, {21290, 27546}, {21383, 34076}, {21747, 28310}, {22020, 27398}, {22054, 28173}, {22088, 37561}, {22126, 23361}, {22145, 23585}, {22147, 28235}, {22329, 37854}, {22357, 28211}, {23073, 28233}, {23165, 34986}, {23201, 26890}, {23343, 29351}, {23472, 25575}, {23646, 24488}, {23843, 29306}, {24170, 33828}, {24190, 33825}, {24502, 24815}, {24549, 30110}, {24625, 32911}, {24713, 25379}, {24779, 28081}, {24995, 30103}, {25066, 30618}, {26036, 26363}, {26265, 28916}, {27097, 33953}, {28163, 37499}, {28509, 31451}, {29092, 31737}, {29219, 31897}, {29479, 30997}, {30379, 34926}, {31561, 32592}, {31562, 32590}, {31563, 32556}, {31564, 32555}, {32094, 35574}, {32642, 32684}, {32653, 32661}, {32670, 36068}, {32671, 36069}, {32672, 36070}, {32673, 36071}, {32686, 32719}, {33946, 33951}, {34526, 37611}, {35128, 38617}, {36037, 36137}, {37211, 37212}, {37736, 38375}

▸ Functional Relations52 relations

• X₁₀₁ = midpoint of X₂₀ and X₁₅₂

• X₁₀₁ = reflection of X_i in X_j for these (i,j): (4, 118), (103, 3), (150, 116)

• X₁₀₁ = isogonal conjugate of X₅₁₄

• X₁₀₁ = isotomic conjugate of X₃₂₆₁

• X₁₀₁ = complement of X₁₅₀

• X₁₀₁ = anticomplement of X₁₁₆

• X₁₀₁ = X₅₉-Ceva conjugate of X₅₅

• X₁₀₁ = cevapoint of X₃₅₄ and X₅₁₃

• X₁₀₁ = X_i-cross conjugate of X_j for these (i,j): (55, 59), (199, 250)

• X₁₀₁ = crosssum of X_i and X_j for these (i,j): (513, 650), (523, 661), (649, 1459)

• X₁₀₁ = crossdifference of every pair of points on line X₁₁X₂₄₄

• X₁₀₁ = X_i-aleph conjugate of X_j for these (i,j): (100, 165), (509, 1052), (662, 572), (664, 169)

• X₁₀₁ = X_i-beth conjugate of X_j for these (i,j): (21, 105), (644, 644)

• X₁₀₁ = circumcircle-antipode of X₁₀₃

• X₁₀₁ = Ψ(X_i,X_j) for these (i,j): (1, 6), (2, 1), (3, 48), (4, 9), (6, 31), (7, 2), (8, 9), (9, 55), (57, 56), (63, 3), (69, 63), (76, 10)

• X₁₀₁ = X₁₁₄-of-the-hexyl-triangle

• X₁₀₁ = trilinear product of PU(i) for these i: 26, 49

• X₁₀₁ = barycentric product of PU(33)

• X₁₀₁ = the point of intersection, other than A, B, C, of the circumcircle and hyperbola {A, B, C,PU(9)}}

• X₁₀₁ = the point of intersection, other than A, B, C, of conic {A, B, C,X₁,PU(93)}}

• X₁₀₁ = trilinear pole of line X₆X₃₁ (the isogonal conjugate of the isotomic conjugate of the Nagel line)

• X₁₀₁ = trilinear pole wrt 1st circumperp triangle of line X₉X₁₆₅

• X₁₀₁ = X₉₉-of -1st-circumperp-triangle

• X₁₀₁ = crossdifference of PU(i) for these i: 121, 123

• X₁₀₁ = concurrence of reflections of line X₄X₉ in sides of ABC

• X₁₀₁ = isogonal conjugate of isotomic conjugate of trilinear pole of Nagel line

• X₁₀₁ = center of Kiepert hyperbola of excentral triangle (i.e. X₁₁₅ of excentral triangle)

• X₁₀₁ = reflection of X₂₆₉₀ in the Euler line

• X₁₀₁ = reflection of X₂₇₀₂ in the Brocard axis

• X₁₀₁ = reflection of X₁₃₀₈ in line X₁X₃

• X₁₀₁ = reflection of X₅₀₁₁ in antiorthic axis

• X₁₀₁ = inverse-in-polar-circle of X₅₁₉₀

• X₁₀₁ = inverse-in-{circumcircle, nine-point circle}-inverter of X₅₅₁₃

• X₁₀₁ = X₆-isoconjugate of X₆₉₃

• X₁₀₁ = X₉₂-isoconjugate of X₁₄₅₉

• X₁₀₁ = X₁₅₇₇-isoconjugate of X₅₈

• X₁₀₁ = eigencenter of 2nd circumperp triangle

• X₁₀₁ = perspector of 3rd mixtilinear triangle and unary cofactor triangle of 5th mixtilinear triangle

• X₁₀₁ = trilinear product of vertices of 1st circumperp triangle

• X₁₀₁ = Thomson-isogonal conjugate of X₅₁₆

• X₁₀₁ = Lucas-isogonal conjugate of X₅₁₆

• X₁₀₁ = intersection of Lemoine axes of 1st & 2nd Montesdeoca bisector triangles

• X₁₀₁ = focus of Yff parabola

• X₁₀₁ = polar conjugate of isogonal conjugate of X₃₂₆₅₆

• X₁₀₁ = trilinear product X₆*X₁₀₀

• X₁₀₁ = trilinear product of circumcircle intercepts of line X₁X₆

• X₁₀₁ = barycentric product of circumcircle intercepts of the Nagel line

• X₁₀₁ = inverse-in-Stevanovic-circle of X₂₂₂₂

• X₁₀₁ = polar conjugate of isotomic conjugate of X₁₃₃₁

• X₁₀₁ = X₂-Ceva conjugate of X₃₉₀₂₆

• X₁₀₁ = perspector of hyperbola {A, B, C,X₅₉,X₆₇₇}}

• X₁₀₁ = X₆₃-isoconjugate of X₇₆₄₉

Notes

Let L_A be the reflection of the line X₁X₇ in line BC, and define L_B and L_C cyclically. Let A' = L_B∩L_C, B' = L_C∩L_A, C' = L_A∩L_B. The lines AA', BB', CC' concur in X₁₀₁. (Randy Hutson, 9/23/2011)

Let J_aJ_bJ_c be the excentral triangle. The Brocard axes of BCJ_a, CAJ_b, ABJ_c concur in X₁₀₁. (Randy Hutson, February 10, 2016)

Let P be a point on line X₄X₉ other than X₄. Let A' be the reflection of P in BC, and define B' and C' cyclically. The circumcircles of AA'B', BC'A', CA'B' concur in X₁₀₁. (Randy Hutson, February 10, 2016)

Let Q be a point on the Nagel line 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″ and C″ cyclically. The circumcircles of AB″C″, BC″A″, CA″B″ concur in X₁₀₁. (Randy Hutson, February 10, 2016)

Let A', B', C' be the intersections of the antiorthic axis and lines BC, CA, AB, resp. The circumcircles of AB'C', BC'A', CA'B' concur in X₁₀₁. (Randy Hutson, February 10, 2016)

X₁₀₁ is the perspector of the anticevian triangle of X₁₀₉ and the unary cofactor triangle of the intangents triangle. (Randy Hutson, January 29, 2018)