X₁₀₈

Ψ(CIRCUMCENTER, INCENTER)

X₁₀₈ = anticomplement of X₁₂₃

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

X₁₀₈ (Ψ(CIRCUMCENTER, INCENTER)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1295, 3}

• Related Centers: X₁₂₉₅ (COLLINGS TRANSFORM OF X₁₂₃), X₃ (CIRCUMCENTER)

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

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

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

Search Value (6, 9, 13): \(-0.081760732885\)

▸ Alternate Coordinates1 form

• Trilinears: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = 1/[(b - c)(b + c - a)(b^{2} + c^{2} - a^{2})]\)

▸ Incident Central Lines268 lines

{1, 102}, {2, 123}, {3, 1295}, {4, 11}, {5, 10746}, {6, 10763}, {7, 1013}, {9, 3213}, {12, 451}, {19, 2291}, {20, 34550}, {23, 37798}, {24, 915}, {25, 105}, {28, 225}, {30, 2694}, {33, 57}, {34, 106}, {35, 37289}, {36, 1785}, {40, 207}, {53, 21773}, {55, 196}, {59, 6099}, {63, 2365}, {65, 74}, {73, 3194}, {81, 20122}, {92, 1311}, {98, 1402}, {99, 811}, {100, 653}, {101, 1783}, {107, 2845}, {109, 1020}, {110, 162}, {111, 1880}, {112, 1415}, {154, 34032}, {158, 32706}, {165, 1767}, {171, 29056}, {186, 2687}, {197, 17903}, {198, 1249}, {204, 223}, {222, 3195}, {226, 4183}, {240, 1758}, {241, 28838}, {242, 2726}, {243, 2723}, {273, 675}, {281, 3209}, {318, 404}, {331, 767}, {342, 1005}, {347, 35988}, {378, 1470}, {386, 34456}, {388, 406}, {393, 2178}, {411, 1895}, {429, 961}, {468, 2752}, {475, 7288}, {476, 34922}, {477, 36130}, {511, 2707}, {512, 2714}, {513, 2720}, {514, 2728}, {515, 2733}, {516, 2739}, {517, 2745}, {518, 2750}, {519, 2756}, {520, 2761}, {521, 6081}, {522, 2765}, {523, 2766}, {524, 2767}, {608, 739}, {644, 29163}, {648, 931}, {656, 2762}, {658, 6183}, {664, 1310}, {676, 6087}, {691, 7476}, {692, 15439}, {727, 1395}, {741, 1396}, {840, 1876}, {901, 7012}, {917, 2352}, {919, 6591}, {925, 3658}, {927, 13149}, {929, 17924}, {934, 36118}, {935, 37965}, {944, 8283}, {953, 1319}, {999, 34231}, {1014, 7282}, {1033, 1604}, {1035, 37818}, {1041, 28017}, {1055, 2202}, {1113, 2586}, {1114, 2587}, {1119, 15728}, {1148, 11491}, {1172, 1400}, {1214, 1297}, {1231, 2366}, {1284, 17985}, {1290, 37964}, {1292, 2283}, {1293, 9525}, {1294, 3651}, {1296, 9531}, {1299, 31385}, {1300, 31384}, {1305, 7437}, {1331, 24029}, {1361, 38513}, {1376, 7046}, {1388, 28203}, {1398, 4186}, {1403, 15323}, {1409, 26717}, {1423, 2212}, {1426, 11363}, {1429, 2356}, {1430, 1458}, {1435, 1477}, {1441, 2373}, {1447, 2862}, {1461, 8059}, {1465, 33849}, {1490, 1712}, {1577, 2769}, {1598, 15251}, {1621, 37295}, {1745, 7114}, {1750, 7008}, {1753, 15803}, {1763, 15278}, {1784, 2695}, {1824, 28471}, {1827, 38451}, {1828, 38452}, {1844, 15932}, {1847, 2369}, {1861, 2751}, {1872, 37582}, {1877, 2718}, {1887, 28173}, {1946, 23984}, {1947, 13588}, {1990, 19297}, {1995, 37800}, {2074, 12030}, {2078, 2717}, {2099, 28159}, {2149, 35182}, {2222, 7649}, {2223, 2724}, {2270, 7129}, {2333, 35106}, {2371, 7079}, {2405, 9107}, {2501, 9090}, {2689, 24006}, {2693, 36001}, {2697, 37959}, {2699, 5061}, {2715, 36104}, {2722, 7178}, {2725, 5236}, {2732, 3465}, {2734, 6905}, {2757, 38462}, {2818, 15501}, {2975, 17555}, {3176, 11500}, {3220, 34050}, {3339, 28149}, {3340, 28163}, {3361, 28193}, {3428, 37410}, {3518, 26707}, {3565, 4236}, {3600, 4194}, {3616, 11733}, {3666, 29206}, {3937, 34051}, {4200, 5265}, {4213, 15168}, {4223, 37695}, {4232, 9061}, {4559, 36080}, {4564, 29241}, {4566, 13395}, {4617, 24016}, {4998, 35574}, {5125, 19850}, {5142, 11392}, {5221, 28145}, {5253, 11109}, {5307, 16878}, {5379, 6083}, {5435, 35994}, {5757, 37538}, {5897, 30267}, {5930, 8885}, {5951, 14882}, {6079, 15742}, {6129, 24033}, {6223, 7338}, {6335, 8707}, {6353, 7337}, {6588, 23985}, {6851, 9645}, {7011, 7580}, {7071, 15731}, {7128, 14733}, {7146, 28844}, {7477, 10420}, {7677, 17923}, {8064, 36049}, {8807, 15324}, {9056, 24035}, {10058, 11798}, {10425, 36105}, {11011, 28185}, {11347, 18678}, {11510, 38295}, {12031, 15148}, {12832, 18341}, {13397, 13589}, {13462, 28233}, {14987, 37117}, {16757, 36093}, {17074, 26910}, {17080, 35996}, {17916, 20613}, {18421, 28171}, {21147, 36103}, {21189, 35183}, {23832, 30236}, {23890, 28291}, {24027, 35187}, {26377, 37384}, {32667, 32683}, {32685, 32702}, {32735, 35185}, {34588, 36100}, {35184, 36109}, {35186, 36112}, {35188, 36115}, {35189, 36116}, {36002, 37769}, {36098, 36099}, {36508, 37055}, {37371, 37797}

▸ Functional Relations29 relations

• X₁₀₈ = reflection of X₁₂₉₅ in X₃

• X₁₀₈ = isogonal conjugate of X₅₂₁

• X₁₀₈ = isotomic conjugate of X₃₅₅₁₈

• X₁₀₈ = complement of X₃₄₁₈₈

• X₁₀₈ = anticomplement of X₁₂₃

• X₁₀₈ = X₁₆₂-Ceva conjugate of X₁₀₉

• X₁₀₈ = cevapoint of X_i and X_j for these (i,j): (56, 513), (429, 523)

• X₁₀₈ = X₅₁₃-cross conjugate of X₄

• X₁₀₈ = crosspoint of X₁₀₇ and X₁₆₂

• X₁₀₈ = crosssum of X₅₂₀ and X₆₅₆

• X₁₀₈ = X_i-beth conjugate of X_j for these (i,j): (21, 102), (162, 108)

• X₁₀₈ = point of intersection, other than A, B, and C, of the circumcircle and hyperbola {A, B, C,PU(18)}}

• X₁₀₈ = trilinear pole of line X₆X₁₉ (the polar of X₄₃₉₁ wrt polar circle)

• X₁₀₈ = pole wrt polar circle of trilinear polar of X₄₃₉₁ (line X₁₁X₁₂₃)

• X₁₀₈ = X₄₈-isoconjugate (polar conjugate) of X₄₃₉₁

• X₁₀₈ = X₁₅₇₇-isoconjugate of X₂₁₉₃

• X₁₀₈ = concurrence of the reflections of line X₄X₆₅ in the sidelines of ABC

• X₁₀₈ = Ψ(X_i,Xj)) for these (i,j): (1, 4), (3, 1), (4, 65), (6, 19), (7, 4), (8, 4), (9, 4), (29, 1), (69, 7), (80, 4)

• X₁₀₈ = reflection of X₂₇₆₆ in the Euler line

• X₁₀₈ = reflection of X₂₇₁₄ in the Brocard axis

• X₁₀₈ = reflection of X₂₇₂₀ in line X₁X₃

• X₁₀₈ = inverse-in-polar-circle of X₁₁

• X₁₀₈ = X₁₃₅-of-excentral-triangle

• X₁₀₈ = barycentric product of PU(76)

• X₁₀₈ = trilinear product of PU(100)

• X₁₀₈ = Thomson-isogonal conjugate of X₆₀₀₁

• X₁₀₈ = Lucas-isogonal conjugate of X₆₀₀₁

• X₁₀₈ = trilinear product of circumcircle intercepts of line X₁X₄

• X₁₀₈ = barycentric product of circumcircle intercepts of line X₂X₉₂