X₃₂₄₆

MIDPOINT OF X₁ AND X₄₄

X₃₂₄₆ is the midpoint of X₁ and X₄₄.

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

X₃₂₄₆ (MIDPOINT OF X₁ AND X₄₄) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 44}

• Related Centers: X₁ (INCENTER), X₄₄ (-LINE CONJUGATE OF X₁)

Trilinears: \(4a^{2} + b^{2} + c^{2} - 4bc - ca - ab : :\)

Barycentrics: \(a(4a^{2} + b^{2} + c^{2} - 4bc - ca - ab) : :\)

Search Value (6, 9, 13): \(-2.352726709109\)

▸ Linear Combinations1 combination

• X₃₂₄₆ = X₁ + \(3\) X₂₃₈, X₁ - \(3\) X₁₂₇₉, \(5\) X₁ + \(3\) X₁₇₅₇, \(5\) X₁ - \(3\) X₄₈₆₄, X₄₄ - \(3\) X₂₃₈, X₄₄ + \(3\) X₁₂₇₉, \(5\) X₄₄ - \(3\) X₁₇₅₇, \(5\) X₄₄ + \(3\) X₄₈₆₄, \(5\) X₂₃₈ - X₁₇₅₇, \(5\) X₂₃₈ + X₄₈₆₄, X₃₂₀ - \(5\) X₃₆₁₆, \(5\) X₁₂₇₉ + X₁₇₅₇, \(5\) X₁₂₇₉ - X₄₈₆₄, \(7\) X₃₆₂₂ + X₂₀₀₇₂, \(7\) X₃₆₂₄ - \(3\) X₃₁₁₅₁, \(7\) X₃₆₂₄ - \(5\) X₃₁₂₄₃, \(4\) X₃₆₃₄ - \(3\) X₃₈₂₃, \(2\) X₃₆₃₄ - \(3\) X₃₁₂₈₉, \(3\) X₃₆₈₅ + X₁₇₁₆₀, \(3\) X₃₈₃₆ - \(5\) X₁₉₈₆₂, \(3\) X₄₆₄₅ - \(11\) X₅₅₅₀, \(7\) X₉₇₈₀ - \(3\) X₃₂₈₅₀, \(3\) X₂₅₀₅₅ - X₃₁₁₃₈, \(3\) X₃₁₁₅₁ - \(5\) X₃₁₂₄₃

▸ Incident Central Lines107 lines

{1, 6}, {8, 17342}, {10, 6687}, {31, 3742}, {36, 16686}, {63, 4906}, {88, 105}, {89, 30653}, {171, 3848}, {239, 4702}, {320, 3616}, {354, 17127}, {513, 1960}, {516, 15251}, {519, 4422}, {528, 3008}, {536, 4432}, {537, 4759}, {551, 4364}, {595, 3812}, {614, 4640}, {651, 1319}, {678, 899}, {748, 3740}, {752, 1125}, {896, 3999}, {902, 16610}, {1086, 28534}, {1149, 3248}, {1193, 27637}, {1443, 1456}, {1621, 17012}, {2239, 30950}, {2243, 3290}, {2308, 4883}, {2325, 28503}, {2999, 4428}, {3011, 5087}, {3052, 5272}, {3058, 26723}, {3271, 9037}, {3416, 29579}, {3550, 16602}, {3617, 17358}, {3622, 20072}, {3624, 31151}, {3626, 17765}, {3634, 3823}, {3683, 7191}, {3685, 17160}, {3689, 37680}, {3696, 16816}, {3745, 5284}, {3748, 32911}, {3749, 37679}, {3752, 8616}, {3791, 4891}, {3836, 19862}, {3844, 3883}, {3880, 3939}, {3912, 28538}, {3915, 5836}, {4009, 20045}, {4346, 5698}, {4395, 28580}, {4421, 23511}, {4423, 4682}, {4645, 5550}, {4660, 17356}, {4670, 24331}, {4679, 26228}, {4719, 5248}, {4849, 17715}, {4860, 36277}, {4887, 17768}, {4896, 25557}, {4914, 33157}, {4974, 28581}, {5211, 26070}, {5263, 16815}, {5269, 8167}, {5308, 38025}, {5880, 16020}, {7262, 21342}, {8301, 9324}, {8583, 25891}, {9780, 17371}, {13624, 15310}, {14997, 21870}, {15808, 28570}, {16694, 20470}, {17013, 37593}, {17237, 29660}, {17325, 25055}, {17335, 36534}, {17348, 32941}, {17357, 33076}, {17450, 21747}, {17605, 29681}, {21214, 28362}, {25351, 28562}, {26626, 38023}, {28352, 28375}, {28370, 28395}, {28555, 32922}, {30148, 31445}, {36289, 37600}

▸ Functional Relations9 relations

• X₃₂₄₆ = midpoint of X_i and X_j for these {i,j}: {1, 44}, {238, 1279}, {239, 4702}, {1155, 14190}, {1757, 4864}

• X₃₂₄₆ = reflection of X_i in X_j for these {i,j}: {10, 6687}, {3823, 31289}, {3834, 1125}

• X₃₂₄₆ = X₅₁₄-isoconjugate of X₆₀₁₇

• X₃₂₄₆ = crosspoint of X₁₀₁₆ and X₉₀₈₉

• X₃₂₄₆ = crosssum of X₁₀₁₅ and X₉₀₃₂

• X₃₂₄₆ = crossdifference of every pair of points on line {45, 513}

• X₃₂₄₆ = barycentric product X_i*X_j for these {i,j}: {100, 6009}, {668, 8658}

• X₃₂₄₆ = barycentric quotient X_i/X_j for these {i,j}: {692, 6017}, {6009, 693}, {8658, 513}

• X₃₂₄₆ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {1, 238, 44}, {1, 15601, 5220}, {44, 1279, 1}, {748, 3744, 3740}, {1001, 1386, 15569}, {1001, 7290, 1386}, {3624, 31151, 31243}, {16672, 38315, 1}