X₄₂₈

EULER X₃₈-5th-SUBSTITUTION POINT

Let H = X₄ and let H_aH_bH_c be the pedal triangle of H in the plane of a triangle ABC. Let

• B_c = reflection of H_b in HH_c, and define C_a and A_b cyclically;

• C_b = reflection of H_c in HH_b, and define A_c and B_a cyclically;

• H_a = orthogonal projecton of A on B_cC_b, and define H_b and H_c cyclically.

Then X₄₂₈ = centroid of triangle H_aH_bH_c. See Antreas Hatzipolakis and Peter Moses, Hyacinthos 27169

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

X₄₂₈ (EULER X₃₈-5th-SUBSTITUTION POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {132, 137}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₁₃₂ (∩X₄X₃₂), X₁₃₇ (-OF-ORTHIC-TRIANGLE)

Trilinears: \(3 \sec A - \csc A \tan \omega : :\)

Barycentrics: \((a^{2} + b^{2} - c^{2}) \cdot (a^{2} - b^{2} + c^{2}) \cdot (2 a^{2} + b^{2} + c^{2}) : :\)

Search Value (6, 9, 13): \(-2.769479717909\)

▸ Alternate Coordinates1 form

• Trilinears: \(\csc A - 3 \sec A \cot \omega : :\)

▸ Linear Combinations1 combination

• X₄₂₈ = \(5\) X₂ - \(4\) X₇₇₃₄, \(3\) X₂ - \(4\) X₁₀₁₂₈, \(7\) X₂ - \(8\) X₁₃₃₆₁, \(4\) X₂ - \(3\) X₄₃₉₅₇, \(4\) X₄ - X₁₈₈₅, \(2\) X₄ + X₃₅₇₅, \(5\) X₄ + X₆₂₄₀, X₄ + \(2\) X₆₇₅₆, \(5\) X₄ - \(2\) X₁₃₄₈₈, \(3\) X₄ + X₁₈₅₅₉, \(7\) X₄ - X₁₈₅₆₀, \(11\) X₄ + X₃₄₇₉₇, \(2\) X₅ + X₇₅₅₃, X₂₀ - \(4\) X₉₈₂₅, \(4\) X₁₄₀ - \(3\) X₄₃₉₃₄, X₃₈₂ + \(2\) X₃₁₈₃₃, \(2\) X₅₄₆ + X₁₁₈₁₉, \(4\) X₅₄₆ - X₁₂₆₀₅, X₅₄₈ - \(4\) X₁₃₁₆₃, \(5\) X₁₆₅₆ - \(8\) X₂₃₄₁₁, X₁₈₈₅ + \(2\) X₃₅₇₅, \(5\) X₁₈₈₅ + \(4\) X₆₂₄₀, X₁₈₈₅ + \(8\) X₆₇₅₆, X₁₈₈₅ + \(4\) X₇₅₇₆, \(5\) X₁₈₈₅ - \(8\) X₁₃₄₈₈, \(3\) X₁₈₈₅ + \(4\) X₁₈₅₅₉, \(7\) X₁₈₈₅ - \(4\) X₁₈₅₆₀, \(11\) X₁₈₈₅ + \(4\) X₃₄₇₉₇, \(5\) X₃₀₉₁ - \(2\) X₁₂₃₆₂, X₃₁₄₆ + \(2\) X₃₁₈₂₉, \(5\) X₃₅₇₅ - \(2\) X₆₂₄₀, X₃₅₇₅ - \(4\) X₆₇₅₆, \(5\) X₃₅₇₅ + \(4\) X₁₃₄₈₈, \(3\) X₃₅₇₅ - \(2\) X₁₈₅₅₉, \(7\) X₃₅₇₅ + \(2\) X₁₈₅₆₀, \(11\) X₃₅₇₅ - \(2\) X₃₄₇₉₇, X₃₆₂₇ + \(2\) X₃₁₈₃₀, \(7\) X₃₈₃₂ - X₁₂₂₂₅, \(3\) X₃₈₃₉ - X₅₂₀₆₉, \(5\) X₃₈₄₃ - X₁₈₅₆₄, \(5\) X₃₈₅₈ - \(2\) X₅₂₀₇₃, \(4\) X₃₈₆₁ - X₅₂₀₇₀, X₆₂₄₀ - \(10\) X₆₇₅₆, X₆₂₄₀ - \(5\) X₇₅₇₆, X₆₂₄₀ + \(2\) X₁₃₄₈₈, \(3\) X₆₂₄₀ - \(5\) X₁₈₅₅₉, \(7\) X₆₂₄₀ + \(5\) X₁₈₅₆₀, \(11\) X₆₂₄₀ - \(5\) X₃₄₇₉₇, and many others

▸ Incident Central Lines265 lines

{1, 51719}, {2, 3}, {6, 31166}, {11, 5322}, {12, 5310}, {19, 5101}, {33, 3058}, {34, 5434}, {51, 1503}, {53, 2980}, {98, 39284}, {107, 29316}, {125, 44106}, {132, 137}, {136, 44953}, {184, 5480}, {185, 11745}, {230, 1879}, {232, 5421}, {242, 7140}, {251, 5305}, {264, 37671}, {275, 14492}, {305, 32819}, {316, 45201}, {317, 7788}, {325, 16276}, {343, 3818}, {389, 16655}, {394, 31670}, {395, 10642}, {396, 10641}, {511, 22480}, {519, 1829}, {524, 1843}, {528, 1824}, {529, 1828}, {530, 12142}, {531, 12141}, {532, 22482}, {533, 22481}, {535, 1878}, {538, 12143}, {539, 5446}, {541, 12133}, {542, 1112}, {543, 5186}, {544, 5185}, {545, 24814}, {551, 11363}, {553, 1876}, {591, 45400}, {597, 1974}, {599, 7716}, {612, 6284}, {614, 7354}, {754, 12144}, {1162, 1163}, {1184, 7737}, {1192, 43903}, {1194, 7745}, {1196, 7747}, {1352, 33586}, {1495, 23292}, {1560, 45161}, {1629, 6530}, {1839, 5089}, {1848, 49736}, {1861, 2355}, {1892, 4654}, {1899, 17810}, {1902, 28194}, {1986, 11566}, {1991, 45401}, {1992, 12167}, {1993, 21850}, {1994, 46818}, {2052, 14458}, {2501, 25423}, {2969, 7009}, {3060, 3564}, {3087, 45141}, {3199, 7753}, {3241, 7718}, {3563, 20189}, {3567, 16659}, {3574, 16252}, {3589, 22352}, {3614, 7302}, {3679, 5090}, {3796, 14561}, {3819, 29317}, {3849, 5140}, {3917, 29181}, {3920, 15171}, {4421, 11383}, {4995, 52427}, {5012, 18583}, {5095, 41149}, {5130, 34606}, {5272, 10483}, {5309, 27376}, {5342, 50310}, {5345, 7741}, {5359, 18907}, {5370, 7173}, {5410, 19054}, {5411, 19053}, {5412, 32787}, {5413, 32788}, {5422, 48906}, {5476, 44077}, {5640, 45298}, {5651, 51163}, {5860, 11389}, {5861, 11388}, {5890, 16658}, {5943, 29012}, {6000, 16654}, {6090, 51538}, {6146, 10110}, {6152, 11817}, {6172, 7717}, {6198, 15170}, {6243, 31831}, {6515, 18440}, {6688, 29323}, {6759, 45089}, {6776, 9777}, {6800, 38136}, {7071, 10385}, {7191, 18990}, {7298, 7951}, {7738, 39951}, {7750, 40022}, {7767, 39998}, {7773, 34254}, {7811, 11386}, {7837, 27377}, {8280, 35786}, {8281, 35787}, {8541, 8584}, {8550, 15004}, {8667, 41762}, {8739, 43229}, {8740, 43228}, {8744, 8792}, {8780, 37645}, {8854, 35821}, {8855, 35820}, {8902, 42400}, {9143, 19504}, {9306, 48901}, {9530, 12145}, {9766, 45478}, {9781, 34224}, {9833, 10982}, {9969, 26926}, {10056, 11398}, {10072, 11399}, {10169, 47459}, {10192, 44082}, {10516, 43653}, {10601, 46264}, {10632, 42912}, {10633, 42913}, {11002, 45968}, {11064, 48895}, {11194, 22479}, {11206, 11402}, {11207, 11384}, {11208, 11385}, {11216, 47463}, {11235, 11390}, {11236, 11391}, {11237, 11392}, {11238, 11393}, {11239, 11400}, {11240, 11401}, {11380, 12150}, {11381, 13568}, {11394, 12152}, {11395, 12153}, {11406, 34607}, {11408, 37640}, {11409, 37641}, {11424, 34782}, {11427, 26864}, {11441, 31802}, {11442, 39884}, {11471, 34618}, {11473, 41945}, {11474, 41946}, {11475, 42942}, {11476, 42943}, {11550, 13567}, {11645, 44084}, {11743, 32332}, {12007, 34565}, {12099, 36201}, {12147, 32419}, {12148, 32421}, {12233, 26883}, {12242, 50414}, {12300, 31834}, {13142, 14516}, {13198, 51734}, {13846, 13884}, {13847, 13937}, {14389, 26881}, {14486, 14593}, {14495, 14569}, {14537, 33842}, {14583, 43089}, {14826, 51212}, {15058, 33523}, {15107, 18358}, {15172, 29815}, {15311, 32062}, {16317, 40326}, {16657, 18400}, {17171, 49738}, {17330, 44103}, {17811, 48910}, {17825, 48905}, {17984, 42394}, {18289, 42269}, {18290, 42268}, {18374, 51744}, {18488, 44158}, {18916, 34780}, {19124, 51737}, {19130, 37649}, {20625, 50938}, {21243, 32269}, {21668, 50668}, {22165, 41585}, {26371, 45696}, {26372, 45697}, {26373, 45699}, {26374, 45698}, {26377, 45700}, {26378, 45701}, {26869, 32064}, {26879, 38848}, {26958, 31860}, {29024, 40998}, {32085, 37765}, {32340, 52008}, {33522, 40330}, {34145, 47220}, {34288, 40144}, {35764, 35822}, {35765, 35823}, {37505, 45185}, {37648, 48884}, {37775, 42137}, {37776, 42136}, {39588, 50979}, {41490, 45502}, {41491, 45503}, {42426, 46439}, {43573, 43823}, {43650, 44882}, {51258, 52142}

▸ Functional Relations30 relations

• X₄₂₈ = midpoint of X_i and X_j for these {i,j}: {2, 34603}, {4, 7576}, {376, 34613}, {381, 7540}, {551, 34633}, {3058, 34666}, {3241, 34668}, {3543, 38323}, {3679, 34657}, {3830, 38321}, {4421, 34655}, {5186, 12132}, {5434, 34653}, {5890, 16658}, {7811, 34661}, {9909, 34659}, {11194, 34663}, {12150, 34651}, {15682, 44458}, {15687, 38322}, {38320, 50687}

• X₄₂₈ = reflection of X_i in X_j for these {i,j}: {1, 51719}, {3, 10127}, {6, 51745}, {549, 23410}, {3575, 7576}, {7576, 6756}, {7667, 2}, {10691, 10128}, {11245, 51}, {34614, 376}, {34634, 551}, {34650, 12150}, {34652, 5434}, {34654, 4421}, {34656, 3679}, {34658, 9909}, {34660, 7811}, {34662, 11194}, {34664, 381}, {34665, 3058}, {34667, 3241}, {52397, 10691}

• X₄₂₈ = isogonal conjugate of X₄₁₄₃₅

• X₄₂₈ = complement of X₅₂₃₉₇

• X₄₂₈ = anticomplement of X₁₀₆₉₁

• X₄₂₈ = circumcircle-inverse of X₃₇₉₂₀

• X₄₂₈ = orthocentroidal-circle-inverse of X₅₀₆₄

• X₄₂₈ = polar-circle-inverse of X₅₁₈₉

• X₄₂₈ = orthoptic-circle-of-the-Steiner-inellipse inverse of X₃₇₉₄₃

• X₄₂₈ = polar conjugate of X₁₀₁₅₉

• X₄₂₈ = isotomic conjugate of the isogonal conjugate of X₄₄₀₉₁

• X₄₂₈ = isogonal conjugate of the isotomic conjugate of X₄₄₁₄₂

• X₄₂₈ = polar conjugate of the isotomic conjugate of X₃₅₈₉

• X₄₂₈ = polar conjugate of the isogonal conjugate of X₅₀₀₇

• X₄₂₈ = orthic-isogonal conjugate of X₄₆₀₂₆

• X₄₂₈ = X_i-Ceva conjugate of X_j for these (i,j): {4, 46026}, {41676, 2489}, {42396, 2501}, {44142, 3589}

• X₄₂₈ = X_i-cross conjugate of X_j for these (i,j): {5007, 3589}, {46026, 44142}

• X₄₂₈ = X_i-isoconjugate of X_j for these (i,j): {1, 41435}, {48, 10159}, {63, 3108}, {656, 7953}, {810, 35137}, {4020, 40425}, {4575, 31065}, {31068, 36060}

• X₄₂₈ = X_i-Dao conjugate of X_j for these (i,j): {3, 41435}, {69, 6292}, {136, 31065}, {428, 52397}, {525, 15527}, {1249, 10159}, {1560, 31068}, {2525, 39691}, {3108, 3162}, {3589, 3933}, {4580, 51906}, {7953, 40596}, {35137, 39062}

• X₄₂₈ = cevapoint of X₅₀₀₇ and X₄₄₀₉₁

• X₄₂₈ = crosspoint of X₄ and X₃₂₀₈₅

• X₄₂₈ = crosssum of X₃ and X₃₉₁₇

• X₄₂₈ = crossdifference of every pair of points on line {647, 22121}

• X₄₂₈ = X₂-of-anti-Ara-triangle

• X₄₂₈ = X₃₅₄-of-orthic-triangle if ABC is acute

• X₄₂₈ = X₂ of 3rd pedal triangle of X₄

• X₄₂₈ = {X₂,X₄}-harmonic conjugate of X₅₀₆₄

• X₄₂₈ = barycentric product X_i*X_j for these {i,j}: {4, 3589}, {6, 44142}, {25, 39998}, {76, 44091}, {83, 46026}, {92, 17469}, {264, 5007}, {278, 4030}, {281, 7198}, {286, 21802}, {393, 7767}, {648, 7927}, {1783, 48152}, {1824, 16707}, {1826, 17200}, {1897, 48101}, {2052, 22352}, {2501, 10330}, {6292, 32085}, {6331, 8664}, {11205, 46104}, {28666, 52395}

• X₄₂₈ = barycentric quotient X_i/X_j for these {i,j}: {4, 10159}, {6, 41435}, {25, 3108}, {112, 7953}, {468, 31068}, {648, 35137}, {2501, 31065}, {3589, 69}, {4030, 345}, {5007, 3}, {6292, 3933}, {7198, 348}, {7767, 3926}, {7927, 525}, {8664, 647}, {10330, 4563}, {11205, 3917}, {17200, 17206}, {17469, 63}, {21802, 72}, {22352, 394}, {28666, 7794}, {32085, 40425}, {39998, 305}, {44091, 6}, {44142, 76}, {46026, 141}, {48101, 4025}, {48152, 15413}

• {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {2, 4, 5064}, {2, 3543, 44442}, {2, 5064, 427}, {2, 6995, 7714}, {2, 7667, 43957}, {2, 7714, 25}, {2, 9909, 44210}, {2, 44442, 31152}, {2, 52397, 10691}, {3, 4, 1907}, {3, 6997, 37439}, {4, 20, 11403}, {4, 24, 1595}, {4, 25, 427}, {4, 28, 1883}, {4, 235, 23047}, {4, 427, 52285}, {4, 1594, 16198}, {4, 1596, 10151}, {4, 1598, 235}, {4, 3089, 7507}, {4, 3518, 15559}, {4, 3575, 1885}, {4, 4186, 1904}, {4, 4198, 4214}, {4, 4222, 429}, {4, 5198, 1906}, {4, 5200, 32587}, {4, 6240, 13488}, {4, 6353, 7378}, {4, 6623, 18386}, {4, 6756, 3575}, {4, 6995, 25}, {4, 7487, 1593}, {4, 7714, 2}, {4, 8889, 7409}, {4, 10301, 468}, {4, 10594, 5}, {4, 14004, 430}, {4, 18533, 1597}, {4, 19219, 3127}, {4, 34484, 1594}, {4, 37122, 3}, {4, 37390, 407}, {4, 37395, 37391}, {4, 44803, 44226}, {4, 52291, 32588}, {4, 52294, 403}, {4, 52301, 5094}, {5, 22, 7499}, {5, 7519, 37899}, {5, 17714, 34002}, {5, 37947, 25337}, {20, 7392, 7484}, {22, 7394, 5}, {22, 10594, 25}, {23, 546, 37454}, {23, 5133, 6676}, {23, 37349, 5133}, {25, 427, 468}, {25, 5064, 2}, {25, 5094, 6353}, {25, 6995, 10301}, {25, 7378, 52297}, {25, 7409, 52293}, {25, 11403, 7484}, {25, 37453, 4232}, {53, 10311, 16318}, {235, 23047, 10019}, {378, 37458, 37931}, {381, 9909, 2}, {382, 5020, 1370}, {382, 30739, 47095}, {403, 37969, 468}, {427, 10301, 25}, {427, 52293, 8889}, {427, 52297, 5094}, {468, 52285, 427}, {472, 473, 297}, {546, 6676, 5133}, {546, 11819, 12605}, {547, 21841, 37943}, {858, 13595, 6677}, {1113, 1114, 37920}, {1368, 3627, 7391}, {1368, 7391, 46517}, {1370, 5020, 30739}, {1585, 1586, 11331}, {1594, 34484, 21841}, {1594, 37943, 547}, {1595, 7715, 24}, {1598, 7517, 10594}, {1829, 49542, 12135}, {1907, 37439, 427}, {1995, 3627, 46517}, {1995, 7391, 1368}, {2043, 2044, 7395}, {2072, 7545, 44233}, {2937, 50137, 140}, {3091, 7494, 7539}, {3146, 7398, 7386}, {3518, 15559, 140}, {3567, 16659, 18914}, {3845, 15687, 18566}, {3845, 15809, 5064}, {3845, 44266, 5066}, {3850, 11548, 37353}, {3850, 37910, 7495}, {3861, 37897, 5169}, {4232, 7409, 8889}, {4232, 8889, 37453}, {4232, 52293, 468}, {5000, 5001, 140}, {5094, 6353, 52297}, {5094, 7378, 427}, {5133, 6676, 37454}, {5133, 37349, 546}, {5576, 18378, 13383}, {5899, 37347, 16618}, {6353, 7378, 5094}, {6353, 52297, 468}, {6353, 52301, 25}, {6636, 7533, 37990}, {6636, 37990, 140}, {6995, 7378, 52301}, {6995, 7394, 10594}, {6995, 7408, 4}, {6995, 7500, 37122}, {6997, 7500, 3}, {6997, 37122, 25}, {7378, 52301, 6353}, {7386, 7398, 11284}, {7387, 7528, 7399}, {7392, 11403, 427}, {7394, 7519, 22}, {7396, 40132, 31255}, {7409, 37453, 427}, {7485, 20062, 550}, {7495, 37353, 11548}, {7499, 37899, 22}, {7530, 11818, 15760}, {7530, 15760, 47093}, {7533, 37900, 140}, {7540, 34659, 34603}, {7714, 15809, 37904}, {8744, 34482, 8792}, {8889, 37453, 52293}, {10110, 13419, 6146}, {10128, 10691, 2}, {10691, 52397, 7667}, {11112, 11113, 11354}, {11206, 14853, 11402}, {11550, 34417, 13567}, {11745, 16621, 185}, {12088, 14788, 16197}, {13568, 16656, 11381}, {15687, 44212, 31133}, {15765, 18585, 7403}, {16198, 21841, 1594}, {17810, 36990, 1899}, {17984, 46104, 42394}, {18494, 18535, 4}, {18586, 18587, 14790}, {20405, 20406, 468}, {31133, 44212, 47097}, {33971, 52448, 14569}, {34658, 34664, 7667}, {34664, 44210, 43957}, {37118, 47485, 37935}, {37353, 37913, 7495}, {37900, 37990, 6636}, {37901, 37943, 21284}, {38282, 52284, 52298}, {39884, 41588, 11442}, {42789, 42790, 44832}, {44091, 46026, 3589}, {44266, 47313, 37904}, {44288, 46030, 10297}

Notes

As a point on the Euler line, X₄₂₈ has Shinagawa coefficients \((F,-3E - 3F)\).

Let V = P(4)-Ceva conjugate of U(4)) and W = U(4)-Ceva conjugate of P(4); then V and W lie on the orthocentroidal circle, and with with PU(4) they are the vertices of a cyclic trapezoid. The midpoint of V and W is X₄₂₈. (Randy Hutson, December 26, 2015)