X₁₇₂₅

MIMOSA TRANSFORM OF X₇₄

Let L denote the line having trilinears of X₁₇₂₅ as coefficients. Then L is the line passing through X₃ perpendicular to the Euler line.

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

X₁₇₂₅ (MIMOSA TRANSFORM OF X₇₄) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 21}, {33, 46}

• Related Centers: X₁ (INCENTER), X₂₁ (SCHIFFLER POINT), X₃₃ (PERSPECTOR OF THE ORTHIC AND INTANGENTS TRIANGLES), X₄₆ (-CEVA CONJUGATE OF X₁)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = - (\cos A)/x + (\cos B)/y + (\cos C)/z\),\(x : y : z\) = X₇₄

Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)

Search Value (6, 9, 13): \(0.865649956693\)

▸ Alternate Coordinates1 form

• Trilinears: \(1 + \cos(2B) + \cos(2C) : 1 + \cos(2C) + \cos(2A) : 1 + \cos(2A) + \cos(2B)\) (Randy Hutson, 9/23/2011)

▸ Incident Central Lines71 lines

{1, 21}, {12, 35194}, {28, 1710}, {33, 46}, {34, 90}, {35, 201}, {36, 5494}, {40, 1775}, {57, 1774}, {79, 45924}, {80, 38945}, {91, 158}, {109, 1727}, {240, 522}, {244, 3582}, {484, 40577}, {495, 24431}, {499, 17063}, {500, 17637}, {564, 45225}, {612, 17699}, {756, 3584}, {912, 8758}, {942, 8143}, {982, 10072}, {984, 10056}, {1172, 1744}, {1210, 36250}, {1214, 4337}, {1254, 3585}, {1393, 7741}, {1399, 18447}, {1406, 7072}, {1454, 37696}, {1464, 2771}, {1478, 24430}, {1479, 37591}, {1755, 3708}, {1758, 3465}, {1776, 1870}, {1779, 5902}, {1807, 5172}, {1858, 37565}, {1973, 21374}, {2083, 2172}, {2159, 2173}, {2166, 18486}, {2181, 17890}, {2247, 9406}, {2310, 3583}, {2312, 16562}, {2349, 36083}, {2361, 18455}, {3649, 5492}, {3670, 24210}, {3911, 16869}, {5348, 37729}, {5497, 32760}, {6198, 7098}, {7069, 7951}, {7073, 45923}, {7082, 37697}, {7100, 8614}, {17757, 24433}, {17879, 18694}, {17881, 40703}, {18180, 42440}, {23580, 26013}, {24028, 41684}, {36034, 36053}, {38336, 41697}

▸ Functional Relations11 relations

• X₁₇₂₅ = reflection of X₂₃₅₈₀ in X₂₆₀₁₃

• X₁₇₂₅ = isogonal conjugate of X₃₆₀₅₃

• X₁₇₂₅ = polar conjugate of the isogonal conjugate of X₂₃₁₅

• X₁₇₂₅ = X_i-Ceva conjugate of X_j for these (i,j): {18609, 3003}, {32680, 661}, {36034, 656}, {36119, 1}

• X₁₇₂₅ = X_i-isoconjugate of X_j for these (i,j): {1, 36053}, {2, 14910}, {3, 1300}, {4, 5504}, {6, 2986}, {30, 10419}, {32, 40832}, {50, 40427}, {74, 15454}, {110, 15328}, {112, 15421}, {113, 39379}, {115, 18879}, {186, 12028}, {254, 15478}, {265, 38936}, {476, 15470}, {477, 39986}, {512, 18878}, {523, 10420}, {525, 32708}, {647, 687}, {656, 36114}, {1495, 40423}, {2501, 43755}, {5627, 39371}, {10733, 39372}, {11064, 40388}, {12383, 35373}, {14385, 39375}, {14911, 34178}, {18315, 35361}

• X₁₇₂₅ = crosspoint of X_i and X_j for these (i,j): {1, 2166}, {75, 2349}

• X₁₇₂₅ = crosssum of X_i and X_j for these (i,j): {1, 6149}, {31, 2173}, {758, 25440}, {2631, 2643}

• X₁₇₂₅ = crossdifference of every pair of points on line {48, 661}

• X₁₇₂₅ = barycentric product X_i*X_j for these {i,j}: {1, 3580}, {10, 18609}, {48, 44138}, {63, 403}, {75, 3003}, {92, 13754}, {113, 2349}, {162, 6334}, {264, 2315}, {304, 44084}, {656, 16237}, {686, 811}, {799, 21731}, {1577, 15329}, {2166, 34834}, {14206, 14264}, {32679, 41512}

• X₁₇₂₅ = barycentric quotient X_i/X_j for these {i,j}: {1, 2986}, {6, 36053}, {19, 1300}, {31, 14910}, {48, 5504}, {75, 40832}, {112, 36114}, {113, 14206}, {162, 687}, {163, 10420}, {403, 92}, {656, 15421}, {661, 15328}, {662, 18878}, {686, 656}, {1101, 18879}, {2159, 10419}, {2166, 40427}, {2173, 15454}, {2315, 3}, {2349, 40423}, {2624, 15470}, {3003, 1}, {3580, 75}, {4575, 43755}, {6334, 14208}, {12824, 16568}, {12827, 20884}, {13754, 63}, {14264, 2349}, {15329, 662}, {16237, 811}, {18609, 86}, {21731, 661}, {32676, 32708}, {39985, 36102}, {41512, 32680}, {44084, 19}, {44138, 1969}

• X₁₇₂₅ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {1, 920, 47}, {1, 1749, 6149}, {31, 18477, 1}, {774, 44706, 1}, {942, 8143, 11553}, {1406, 7072, 18451}, {1735, 1736, 1737}, {1735, 1737, 1772}, {1749, 6149, 896}, {1755, 3708, 18669}, {1822, 1823, 47}, {3743, 10122, 1}