X₁₀₉

Ψ(INCENTER, CIRCUMCENTER)

X₁₀₉ = anticomplement of X₁₂₄

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

X₁₀₉ (Ψ(INCENTER, CIRCUMCENTER)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {20, 151}

• Related Centers: X₂₀ (DE LONGCHAMPS POINT), X₁₅₁ (REFLECTION OF X₂₀ IN X₁₀₉)

Trilinears: \(a/(\cos B - \cos C) : b/(\cos C - \cos A) : c/(\cos A - \cos B)\)

Barycentrics: \(a^{2}/(\cos B - \cos C) : b^{2}/(\cos C - \cos A): c^{2}/(\cos A - \cos B)\)

Tripolars: \(|b c (\cos B - \cos C)| : :\)

Search Value (6, 9, 13): \(0.160926157179\)

▸ Alternate Coordinates2 forms

• Trilinears: \(a/[(b - c)(b + c - a)] : :\)

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

▸ Incident Central Lines413 lines

{1, 104}, {2, 124}, {3, 102}, {4, 117}, {5, 10747}, {6, 2291}, {7, 675}, {8, 2370}, {9, 2199}, {10, 1935}, {11, 10777}, {12, 2372}, {19, 8775}, {20, 151}, {21, 37558}, {30, 2695}, {31, 57}, {33, 1709}, {34, 46}, {35, 73}, {36, 953}, {37, 24863}, {40, 255}, {42, 2003}, {48, 32726}, {55, 103}, {56, 106}, {58, 65}, {59, 901}, {63, 8270}, {77, 35258}, {79, 1141}, {85, 767}, {98, 171}, {99, 643}, {100, 651}, {101, 654}, {107, 162}, {108, 1020}, {110, 1813}, {111, 1400}, {112, 163}, {140, 38776}, {154, 7011}, {165, 212}, {181, 20670}, {187, 17966}, {190, 6648}, {191, 201}, {204, 1767}, {213, 35106}, {220, 2371}, {225, 1300}, {227, 3579}, {238, 2726}, {241, 2725}, {260, 7597}, {269, 15728}, {278, 917}, {279, 2369}, {283, 4296}, {284, 296}, {307, 2373}, {349, 2367}, {376, 38783}, {381, 38780}, {386, 11509}, {388, 5264}, {394, 2365}, {476, 13486}, {477, 15228}, {478, 573}, {484, 2687}, {511, 2708}, {512, 2701}, {513, 2222}, {514, 929}, {515, 2734}, {516, 1936}, {517, 1455}, {518, 2751}, {519, 2757}, {520, 2762}, {521, 2765}, {522, 1309}, {523, 2689}, {524, 2768}, {525, 2769}, {549, 38786}, {550, 38778}, {553, 9108}, {572, 29068}, {577, 1630}, {579, 608}, {581, 11507}, {602, 15803}, {604, 739}, {631, 6711}, {644, 6574}, {649, 919}, {652, 7115}, {656, 2766}, {658, 927}, {660, 8684}, {661, 9090}, {662, 931}, {663, 1262}, {677, 4105}, {689, 4572}, {692, 1461}, {727, 1397}, {738, 2377}, {741, 1402}, {748, 31231}, {750, 5219}, {753, 1469}, {761, 7146}, {765, 6079}, {789, 4554}, {810, 2714}, {835, 4427}, {840, 902}, {846, 15168}, {896, 2752}, {898, 4564}, {905, 2728}, {932, 3573}, {933, 36134}, {934, 4617}, {940, 29310}, {946, 3075}, {952, 18339}, {958, 3040}, {959, 34281}, {993, 24806}, {995, 1470}, {999, 15306}, {1026, 4571}, {1035, 2122}, {1038, 12514}, {1042, 37583}, {1046, 15556}, {1086, 15253}, {1106, 1420}, {1110, 2742}, {1113, 1822}, {1114, 1823}, {1149, 5193}, {1155, 1456}, {1174, 20229}, {1201, 38452}, {1210, 3073}, {1214, 2328}, {1252, 6078}, {1253, 1419}, {1279, 3660}, {1284, 5061}, {1290, 4017}, {1292, 2814}, {1293, 2815}, {1294, 2816}, {1296, 2819}, {1297, 5285}, {1304, 36034}, {1305, 4566}, {1308, 3669}, {1310, 6516}, {1319, 2718}, {1365, 24836}, {1376, 3042}, {1393, 2964}, {1395, 1707}, {1401, 28485}, {1404, 2384}, {1405, 28317}, {1406, 4306}, {1407, 1477}, {1413, 10306}, {1423, 9082}, {1425, 3145}, {1428, 2382}, {1429, 14665}, {1434, 2368}, {1435, 2376}, {1436, 22124}, {1445, 9061}, {1447, 9073}, {1448, 37550}, {1450, 5315}, {1451, 3339}, {1459, 1618}, {1460, 20678}, {1464, 5172}, {1466, 16466}, {1468, 3340}, {1490, 2956}, {1492, 30670}, {1496, 1697}, {1497, 3333}, {1498, 12330}, {1499, 2735}, {1621, 17074}, {1656, 38779}, {1702, 3076}, {1703, 3077}, {1710, 1825}, {1724, 1788}, {1725, 1727}, {1735, 1870}, {1736, 1776}, {1737, 1877}, {1745, 6796}, {1758, 5018}, {1764, 34045}, {1780, 7098}, {1783, 35349}, {1794, 3466}, {1807, 2771}, {1818, 2750}, {1836, 5348}, {1918, 6015}, {1923, 3503}, {1943, 32932}, {1955, 2655}, {1995, 34141}, {2006, 3120}, {2077, 2745}, {2177, 28535}, {2187, 7125}, {2192, 38288}, {2223, 12032}, {2285, 4264}, {2293, 38451}, {2352, 29015}, {2360, 22341}, {2406, 9056}, {2425, 26715}, {2594, 8614}, {2605, 34921}, {2617, 3658}, {2688, 6357}, {2690, 7178}, {2699, 5143}, {2700, 17798}, {2702, 7180}, {2724, 9441}, {2729, 7181}, {2730, 3309}, {2731, 3667}, {2732, 6000}, {2733, 6001}, {2737, 30719}, {2739, 9371}, {2756, 5440}, {2760, 3712}, {2810, 22148}, {2829, 18340}, {2860, 24002}, {2861, 22464}, {2862, 9436}, {2864, 4077}, {2866, 3263}, {3028, 13868}, {3033, 20804}, {3072, 4292}, {3074, 6684}, {3090, 38781}, {3157, 11248}, {3173, 3190}, {3197, 15905}, {3218, 4318}, {3241, 14648}, {3295, 28193}, {3303, 28227}, {3451, 20228}, {3476, 37610}, {3485, 37522}, {3523, 38784}, {3533, 38782}, {3550, 15323}, {3563, 36051}, {3576, 11713}, {3585, 18426}, {3616, 11734}, {3652, 35194}, {3659, 6733}, {3699, 8706}, {3722, 37736}, {3744, 17625}, {3746, 28173}, {3750, 29308}, {3937, 20999}, {3952, 9059}, {3955, 29056}, {4220, 34027}, {4295, 37530}, {4303, 10902}, {4347, 37591}, {4418, 6358}, {4557, 8694}, {4569, 34083}, {4570, 6083}, {4587, 29163}, {4620, 9150}, {4654, 9103}, {4848, 5247}, {4998, 8709}, {5010, 28159}, {5089, 8558}, {5127, 12030}, {5204, 28235}, {5217, 28163}, {5248, 37523}, {5255, 10106}, {5297, 29007}, {5398, 36279}, {5399, 11849}, {5435, 9083}, {5537, 34143}, {5563, 28219}, {5603, 11727}, {5687, 9370}, {5840, 10771}, {5880, 37695}, {5897, 8803}, {5903, 14987}, {6014, 8683}, {6180, 29352}, {6244, 7074}, {6327, 28774}, {6589, 23979}, {6610, 19624}, {6745, 23693}, {6759, 20764}, {6788, 12832}, {7070, 10860}, {7179, 9075}, {7186, 29096}, {7248, 28574}, {7280, 28203}, {7292, 37789}, {7299, 24914}, {7339, 24016}, {7411, 34035}, {7580, 34032}, {7589, 34026}, {7676, 34028}, {7991, 34039}, {8059, 32652}, {8075, 34025}, {8076, 34034}, {8107, 34037}, {8108, 34038}, {8224, 34031}, {8543, 37633}, {8679, 34142}, {8757, 11499}, {9058, 24029}, {9095, 30653}, {9105, 21454}, {9363, 37588}, {9778, 18623}, {10303, 38787}, {10434, 34044}, {10700, 20586}, {11246, 26708}, {12709, 35672}, {13138, 31511}, {13589, 33637}, {13597, 37731}, {14074, 23890}, {14107, 37527}, {14115, 37815}, {14628, 24402}, {14647, 34231}, {14827, 20995}, {15252, 38357}, {15386, 35183}, {15439, 32651}, {15931, 22053}, {16173, 29008}, {18838, 30117}, {19369, 28559}, {22097, 29206}, {22329, 37856}, {22456, 36036}, {23071, 35000}, {23850, 30493}, {23987, 26704}, {24026, 24410}, {24033, 36044}, {25440, 37694}, {25577, 28847}, {25882, 36949}, {25938, 30827}, {25968, 26932}, {26227, 28968}, {26264, 29497}, {26740, 29821}, {28148, 35327}, {28777, 29473}, {28780, 33086}, {28844, 37586}, {28848, 37580}, {32643, 32683}, {32669, 32685}, {32675, 34073}, {32687, 36046}, {34234, 34589}, {34789, 35015}, {35448, 37483}, {36040, 36067}, {36048, 36118}

▸ Functional Relations40 relations

• X₁₀₉ = midpoint of X₂₀ and X₁₅₁

• X₁₀₉ = reflection of X_i in X_j for these (i,j): (4, 117), (102, 3)

• X₁₀₉ = isogonal conjugate of X₅₂₂

• X₁₀₉ = complement of X₃₃₆₅₀

• X₁₀₉ = anticomplement of X₁₂₄

• X₁₀₉ = X_i-Ceva conjugate of X_j for these (i,j): (59, 56), (162, 108)

• X₁₀₉ = cevapoint of X₆₅ and X₅₁₃

• X₁₀₉ = X_i-cross conjugate of X_j for these (i,j): (56, 59), (513, 58)

• X₁₀₉ = crosspoint of X₁₁₀ and X₁₆₂

• X₁₀₉ = crosssum of X_i and X_j for these (i,j): (523, 656), (652, 663)

• X₁₀₉ = crossdifference of every pair of points on line X₁₁X₁₁₄₆

• X₁₀₉ = X_i-aleph conjugate of X_j for these (i,j): (100, 1079), (162, 580), (651, 223)

• X₁₀₉ = X_i-beth conjugate of X_j for these (i,j): (21, 104), (59, 109), (100, 100), (110, 109), (765, 109), (901, 109)

• X₁₀₉ = circumcircle-antipode of X₁₀₂

• X₁₀₉ = trilinear product X₁₃₈₁*X₁₃₈₂

• X₁₀₉ = isogonal conjugate of isotomic conjugate of trilinear pole of line X₂X₇

• X₁₀₉ = the point of intersection, other than A, B, and C, of the circumcircle and hyperbola {A, B, C,PU(19)}

• X₁₀₉ = trilinear pole of line X₆X₄₁

• X₁₀₉ = trilinear pole wrt 1st circumperp triangle of line X₉₇₁X₁₁₅₈

• X₁₀₉ = trilinear pole, wrt circumtangential triangle, of line X₃X₁₀

• X₁₀₉ = X₉₂₅-of-1st-circumperp-triangle

• X₁₀₉ = X₁₃₆-of-excentral-triangle

• X₁₀₉ = X₁₃₁-of-hexyl-triangle

• X₁₀₉ = Ψ(X_i,X_j) for these (i,j): (1, 3), (2, 7), (3, 73), (4, 1), (6, 41), (7, 20), (21, 2), (69, 73), (77, 3)

• X₁₀₉ = reflection of X₂₆₈₉ in the Euler line

• X₁₀₉ = reflection of X₂₇₀₁ in the Brocard axis

• X₁₀₉ = reflection of X₂₂₂₂ in line X₁X₃

• X₁₀₉ = X₆-isoconjugate of X₄₃₉₁

• X₁₀₉ = X₁₉-isoconjugate of X₆₃₃₂

• X₁₀₉ = X₉₂-isoconjugate of X₆₅₂

• X₁₀₉ = X₁₅₇₇-isoconjugate of X₂₈₄

• X₁₀₉ = barycentric product of PU(57)

• X₁₀₉ = trilinear product of PU(102)

• X₁₀₉ = Thomson isogonal conjugate of X₅₁₅

• X₁₀₉ = Lucas isogonal conjugate of X₅₁₅

• X₁₀₉ = isotomic conjugate of polar conjugate of X₃₂₆₇₄

• X₁₀₉ = polar conjugate of isogonal conjugate of X₃₂₆₆₀

• X₁₀₉ = intersection of antipedal lines of circumcircle intercepts of line X₃X₁₀

• X₁₀₉ = barycentric product of circumcircle intercepts of line X₂X₇

• X₁₀₉ = Conway-circle-inverse of X₃₅₆₄₉

Notes

If the line X₁X₄ is reflected in every side of triangle ABC, then the reflections concur in X₁₀₉. (Randy Hutson, 9/23/2011)

Let P be a point on line X₁X₄ other than X₄. Let A' be the reflection of P in BC, and define B' and C' cyclically. The circumcircles of CA'B', BC'A', and CA'B' concur in X₁₀₉. (Randy Hutson, December 26, 2011)

Let Q be a point on line X₂X₇ 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 X₁₀₉. (Randy Hutson, December 26, 2011)

Let A', B', C' be the Fuhrmann triangle. The circumcircles of AB'C', BC'A', CA'B' concur in X₁₀₉. (Randy Hutson, December 26, 2011)

Let A', B', C' be the intersections of the Gergonne line and lines BC, CA, AB, respectively. The circumcircles of AB'C', BC'A', CA'B' concur in X₁₀₉. (Randy Hutson, December 26, 2011)

Let P and Q be the points where the line tangent to the incircle at X₁₁ intersects the circumcircle. Let L(P) be the line through P, other than PQ, that is tangent to the incircle; let L(Q) be the line through Q, other than PQ, that is tangent to the incircle. Then X₁₀₉ = L(P)∩L(Q). (Piotr Ambroszczyk, December 21, 2016)