X₁₇₄₂

MIMOSA TRANSFORM OF X₂₁₂

X₁₇₄₂ = excentral-isogonal conjugate of X₆₃

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

X₁₇₄₂ (MIMOSA TRANSFORM OF X₂₁₂) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 7}, {43, 165}

• Related Centers: X₁ (INCENTER), X₇ (GERGONNE POINT), X₄₃ (-CEVA CONJUGATE OF X₁), X₁₆₅ (CENTROID OF THE EXCENTRAL TRIANGLE)

Trilinears: \(- (\cos A)/x + (\cos B)/y + (\cos C)/z : :\), where \(x : y : z\) = X₂₁₂

Barycentrics: \(a (a^{3} b - 2 a^{2} b^{2} + a b^{3} + a^{3} c + a^{2} b c - a b^{2} c - b^{3} c - 2 a^{2} c^{2} - a b c^{2} + 2 b^{2} c^{2} + a c^{3} - b c^{3})\) (Peter Moses, April 3, 2020)

Search Value (6, 9, 13): \(5.224682507243\)

▸ Alternate Coordinates1 form

• Trilinears: \(\sec^{2}(B/2) + \sec^{2}(C/2) - \sec^{2}(A/2) : :\) (Randy Hutson, September 23, 2011)

▸ Linear Combinations1 combination

• 3 X₁₆₅ - 2 X₅₇₃, 3 X₁₆₉₉ - 4 X₂₄₂₂₀, 3 X₃₅₇₆ - 2 X₃₁₃₉₄

▸ Incident Central Lines158 lines

{1, 7}, {3, 238}, {6, 9441}, {9, 6184}, {10, 26059}, {31, 7411}, {35, 1745}, {37, 15726}, {38, 11220}, {40, 511}, {42, 9778}, {43, 165}, {45, 16112}, {46, 37409}, {48, 1633}, {55, 6180}, {57, 2942}, {58, 12511}, {75, 28850}, {87, 572}, {103, 7220}, {144, 2340}, {171, 7580}, {200, 4416}, {241, 14100}, {259, 503}, {266, 844}, {371, 31544}, {372, 31545}, {376, 1064}, {386, 12512}, {411, 37603}, {497, 22053}, {500, 37529}, {513, 15624}, {581, 31730}, {601, 3651}, {651, 1253}, {750, 36002}, {846, 1709}, {971, 984}, {982, 10167}, {986, 9943}, {988, 9841}, {1026, 25728}, {1193, 3522}, {1212, 15587}, {1250, 30300}, {1385, 24661}, {1418, 5572}, {1423, 2223}, {1490, 5293}, {1697, 2943}, {1699, 24220}, {1750, 5268}, {1766, 18788}, {1818, 5698}, {1935, 37601}, {1959, 12530}, {2066, 30297}, {2292, 9961}, {2309, 37416}, {2635, 5218}, {2792, 9862}, {2807, 5119}, {2808, 3688}, {2876, 7289}, {3056, 34253}, {3061, 18252}, {3062, 3731}, {3169, 9025}, {3177, 21084}, {3216, 16192}, {3474, 14547}, {3501, 17792}, {3510, 37619}, {3576, 16528}, {3579, 37699}, {3666, 5918}, {3720, 9812}, {3736, 4229}, {3752, 10178}, {3811, 17770}, {3817, 25502}, {3836, 36652}, {3870, 17364}, {3875, 7976}, {3879, 28849}, {3886, 34282}, {3888, 22370}, {3912, 21629}, {3950, 9950}, {4073, 25083}, {4192, 20368}, {4221, 30269}, {4512, 37175}, {4551, 35445}, {4641, 7964}, {5010, 6127}, {5217, 37694}, {5222, 20978}, {5247, 5584}, {5272, 10857}, {5414, 30296}, {5539, 34196}, {5691, 15971}, {5759, 24695}, {5851, 17334}, {5942, 28118}, {6284, 37523}, {6610, 30621}, {6765, 34379}, {7146, 12723}, {7175, 37580}, {7671, 17092}, {7963, 7987}, {7982, 29309}, {7991, 29311}, {8616, 15931}, {8727, 33111}, {8844, 16557}, {9779, 30950}, {9801, 17316}, {10164, 16569}, {10186, 17321}, {10638, 30301}, {10860, 17594}, {10868, 35269}, {11227, 17063}, {12618, 29674}, {13243, 36263}, {13329, 16468}, {15717, 27627}, {16132, 29097}, {17122, 19541}, {17126, 35986}, {17377, 28870}, {17378, 28854}, {17601, 17613}, {17668, 24341}, {17717, 37374}, {20556, 30035}, {20793, 34497}, {20995, 21856}, {21173, 23696}, {22836, 28508}, {24440, 31787}, {24635, 25722}, {26669, 35293}, {28164, 30116}, {29057, 30273}, {30503, 33781}, {31151, 36721}, {31805, 37592}, {35242, 37732}, {35658, 37552}, {37426, 37570}, {37501, 37607}

▸ Functional Relations15 relations

• X₁₇₄₂ = reflection of X_i in X_j for these (i,j): (1, 991), (6210, 3)

• X₁₇₄₂ = isogonal conjugate of the isotomic conjugate of X₂₀₉₃₅

• X₁₇₄₂ = X_i-Ceva conjugate of X_j for these (i,j): (55, 1), (6180, 1743), (31526, 34497)

• X₁₇₄₂ = X_i-cross conjugate of X_j for these (i,j): (20995, 34497), (21856, 3177), (31526, 1)

• X₁₇₄₂ = crosspoint of X_i and X_j for these (i,j): {1, 36601}, {651, 24011}, {3177, 31526}

• X₁₇₄₂ = crosssum of X₆₅₀ and X₂₄₀₁₂

• X₁₇₄₂ = crossdifference of every pair of points on line {657, 4449}

• X₁₇₄₂ = excentral-isogonal conjugate of X₆₃

• X₁₇₄₂ = excentral-isotomic conjugate of X₄₀

• X₁₇₄₂ = X₂₆₄-of-excentral triangle

• X₁₇₄₂ = Brianchon point of the MacBeacth inconic of the excentral triangle

• X₁₇₄₂ = X_i-aleph conjugate of X_j for these (i,j): (1, 57), (6, 978), (9, 40), (55, 1742), (174, 1445), (188, 63), (259, 1), (365, 1743), (366, 169)

• X₁₇₄₂ = barycentric product X_i*X_j for these {i,j}: {1, 3177}, {6, 20935}, {8, 34497}, {9, 31526}, {75, 20995}, {81, 21084}, {86, 21856}, {92, 20793}, {100, 21195}

• X₁₇₄₂ = barycentric quotient X_i/X_j for these {i,j}: {3177, 75}, {20793, 63}, {20935, 76}, {20995, 1}, {21084, 321}, {21195, 693}, {21856, 10}, {31526, 85}, {34497, 7}

• X₁₇₄₂ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {1, 2951, 1721}, {6, 11495, 9441}, {7, 2293, 1}, {20, 4300, 1}, {55, 6180, 9440}, {77, 4319, 1}, {170, 4335, 1721}, {269, 4326, 1}, {390, 1458, 1}, {651, 7676, 1253}, {1042, 4313, 1}, {1442, 4336, 1}, {1716, 1740, 978}, {2263, 7675, 1}, {2293, 3000, 7}, {3945, 4343, 1}, {4294, 4303, 1}, {4302, 4337, 1}, {4306, 4314, 1}, {4322, 9785, 1}, {7671, 17092, 21346}, {9943, 15852, 986}, {30354, 30355, 4326}, {31573, 31574, 12565}