X₃₈₄

THE CONWAY POINT: EULER LINE INTERCEPT OF LINE X₃₂X_76

X₃₈₄ = eigencenter of anticevian triangle of X₃₈₅

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

X₃₈₄ (THE CONWAY POINT: EULER LINE INTERCEPT OF LINE X₃₂X_76) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {32, 76}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₃₂ (3rd POWER POINT), X₇₆ (3rd BROCARD POINT)

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

Barycentrics: \(a^{4} + b^{2}c^{2} : b^{4} + c^{2}a^{2} : c^{4} + a^{2}b^{2}\) (Peter Moses, April 15, 2020)

Search Value (6, 9, 13): \(1.999818533614\)

▸ Linear Combinations2 combinations

• X₃₈₄ = \(3 \csc^{2}(\omega )\) X₂ - \(4\) X₃ = \(3\) X₂ - \(4 \sin^{2}(\omega )\) X₃

• X₃₈₄ = \(3\) X₂ + X₆₆₅₈, \(2\) X₅₀₀₇ - \(3\) X₁₂₁₅₀, X₆₆₅₅ - \(6\) X₆₆₆₁, \(2\) X₆₆₅₆ + X₆₆₅₈, X₆₆₅₆ - \(3\) X₆₆₆₁, X₆₆₅₈ + \(6\) X₆₆₆₁, X₇₇₆₀ - \(3\) X₁₂₁₅₀, \(2\) X₇₇₆₅ - \(3\) X₇₈₂₇, \(3\) X₇₈₂₇ - \(4\) X₇₈₂₉, \(4\) X₇₈₄₉ - \(3\) X₇₈₈₃, \(2\) X₇₈₇₃ - \(3\) X₇₈₈₃, \(3\) X₂₂₅₂₁ - \(4\) X₃₂₁₃₄

X₃₈₄ lies on these lines these conics: {A, B, C,X₄,X₃₈₄,X₁₀₃₁,X₂₉₉₈,X₃₁₁₄,X₉₂₃₀,X₁₄₉₇₀}} and {A, B, C,X₂₅,X₃₈₄,X₇₃₃,X₁₉₁₅,X₃₂₂₄,X₃₄₀₇, X₁₁₃₈₀,X(14370}}

X₃₈₄ lies on these cubics: K020, K322, K421, K532, K828

▸ Incident Central Lines329 lines

{1, 335}, {2, 3}, {6, 194}, {11, 26686}, {12, 26629}, {31, 17033}, {32, 76}, {35, 27020}, {36, 26959}, {37, 20136}, {39, 83}, {41, 24514}, {58, 17034}, {61, 12204}, {62, 12205}, {69, 5017}, {75, 4426}, {98, 6248}, {100, 26752}, {110, 35399}, {112, 1235}, {115, 5152}, {141, 1031}, {148, 5254}, {172, 350}, {182, 10131}, {183, 3053}, {185, 287}, {187, 1078}, {193, 32830}, {218, 17350}, {238, 16827}, {239, 5247}, {251, 1241}, {257, 3496}, {262, 9737}, {264, 1968}, {274, 17000}, {315, 3314}, {316, 626}, {325, 7745}, {330, 16502}, {511, 10333}, {524, 34604}, {538, 5007}, {543, 7765}, {574, 7782}, {575, 5182}, {576, 22486}, {597, 8591}, {598, 7775}, {599, 9939}, {609, 3760}, {620, 1506}, {625, 7874}, {671, 7817}, {694, 695}, {710, 9230}, {730, 12194}, {732, 12206}, {754, 7768}, {904, 6196}, {910, 25994}, {925, 27367}, {958, 20172}, {993, 17030}, {1001, 20133}, {1089, 30167}, {1107, 20179}, {1194, 16276}, {1352, 9863}, {1376, 26687}, {1468, 17027}, {1495, 35301}, {1580, 24259}, {1632, 29959}, {1655, 5276}, {1691, 24256}, {1724, 20142}, {1799, 8891}, {1843, 9229}, {1909, 1914}, {1915, 4074}, {2134, 17398}, {2178, 26107}, {2220, 3770}, {2221, 3210}, {2251, 4721}, {2482, 9698}, {2548, 7763}, {2549, 7803}, {2782, 3398}, {2794, 35385}, {2887, 30175}, {2975, 26801}, {3051, 3499}, {3094, 10345}, {3095, 10334}, {3096, 7761}, {3117, 4159}, {3118, 10337}, {3172, 9308}, {3491, 17970}, {3492, 19558}, {3494, 3502}, {3495, 3503}, {3500, 3501}, {3583, 30103}, {3585, 30104}, {3589, 5116}, {3618, 7738}, {3642, 9988}, {3643, 9989}, {3662, 24549}, {3670, 30111}, {3761, 7031}, {3767, 7806}, {3785, 16990}, {3788, 5475}, {3797, 7283}, {3818, 9873}, {3849, 7849}, {3923, 16822}, {3926, 7774}, {3933, 7762}, {4045, 7756}, {4116, 4172}, {4251, 17499}, {4257, 29455}, {4279, 24267}, {4339, 17316}, {4352, 17379}, {4372, 33931}, {4376, 33930}, {4386, 6376}, {4427, 25248}, {4577, 14885}, {4680, 30149}, {5008, 7805}, {5013, 11174}, {5015, 30179}, {5023, 15271}, {5026, 5038}, {5039, 32451}, {5041, 32450}, {5088, 7187}, {5132, 20148}, {5171, 22712}, {5172, 28771}, {5206, 7771}, {5248, 27255}, {5251, 16819}, {5255, 17752}, {5258, 16829}, {5259, 31996}, {5264, 30114}, {5275, 27269}, {5277, 16999}, {5280, 25264}, {5286, 7920}, {5291, 17143}, {5304, 6392}, {5306, 19570}, {5309, 7856}, {5395, 32831}, {5651, 35275}, {5902, 30139}, {5903, 30136}, {5969, 12191}, {5976, 34870}, {6228, 9992}, {6229, 9991}, {6272, 10793}, {6273, 10792}, {6284, 26590}, {6292, 6781}, {6337, 7736}, {6683, 32456}, {7354, 26561}, {7618, 31450}, {7697, 10104}, {7709, 10359}, {7746, 7857}, {7748, 7790}, {7749, 17006}, {7753, 7764}, {7754, 7766}, {7755, 14568}, {7757, 7772}, {7758, 7837}, {7759, 7796}, {7773, 7778}, {7776, 7881}, {7780, 9466}, {7784, 7868}, {7788, 7946}, {7798, 7894}, {7800, 7904}, {7809, 7821}, {7811, 7854}, {7813, 7838}, {7818, 7860}, {7825, 7867}, {7842, 7853}, {7844, 7942}, {7845, 7895}, {7848, 32027}, {7850, 7896}, {7851, 7932}, {7852, 7861}, {7855, 7877}, {7862, 7940}, {7865, 7936}, {7871, 7903}, {7872, 7913}, {7879, 7929}, {7884, 7902}, {7886, 14061}, {7910, 7914}, {7916, 7949}, {7976, 10800}, {8267, 34482}, {8667, 22331}, {8782, 10346}, {8790, 19573}, {8868, 8875}, {8870, 14509}, {8992, 13885}, {9301, 32521}, {9605, 31859}, {9766, 32821}, {9902, 10789}, {9917, 10790}, {10027, 37588}, {10063, 10801}, {10079, 10802}, {10159, 31168}, {10313, 26214}, {10353, 13188}, {10788, 12251}, {10791, 12782}, {10794, 12923}, {10795, 12933}, {10797, 12837}, {10798, 12836}, {10799, 13077}, {10803, 13109}, {10804, 13110}, {10958, 28925}, {10992, 25555}, {11164, 32480}, {11272, 33813}, {11364, 12263}, {11380, 12143}, {11490, 12338}, {11638, 19663}, {11837, 12474}, {11838, 12475}, {11839, 12794}, {11840, 12992}, {11841, 12993}, {11842, 13108}, {12122, 14810}, {12195, 14839}, {12835, 18982}, {13030, 13033}, {13032, 13034}, {13232, 13237}, {13571, 32820}, {13684, 13686}, {13804, 13806}, {13938, 13983}, {14135, 34236}, {14377, 24190}, {14822, 20027}, {14880, 26316}, {14881, 18502}, {14962, 27375}, {15048, 20094}, {15482, 15515}, {15513, 31239}, {16275, 21248}, {16500, 20144}, {16589, 16993}, {16689, 18091}, {16783, 20147}, {16946, 34283}, {16991, 26085}, {16997, 18135}, {16998, 34284}, {17004, 32832}, {17008, 32828}, {17103, 24512}, {17729, 24170}, {18393, 30120}, {18755, 37678}, {18800, 33749}, {18899, 33786}, {18993, 19089}, {18994, 19090}, {19761, 19768}, {20105, 22253}, {20180, 20963}, {21850, 35458}, {22398, 36511}, {22520, 22779}, {22521, 32134}, {22677, 22679}, {22736, 22748}, {22737, 22749}, {22760, 28934}, {24206, 32152}, {24279, 24502}, {24358, 33943}, {24726, 25364}, {25406, 35423}, {25440, 27091}, {25957, 30174}, {26068, 32561}, {26164, 30737}, {26613, 34506}, {29012, 35422}, {30135, 37571}, {30140, 37525}, {31404, 32829}, {32834, 37667}, {33863, 37686}, {35766, 35866}, {35767, 35867}

▸ Functional Relations43 relations

• X₃₈₄ = midpoint of X₂ and X₁₉₆₈₆

• X₃₈₄ = reflection of X_i in X_j for these {i,j}: (2, 6661), (6655, 6656), (7760, 5007), (7765, 7829), (7768, 7794), (7873, 7849), (24726, 25364)

• X₃₈₄ = isogonal conjugate of X₆₉₅

• X₃₈₄ = isotomic conjugate of X₉₂₂₉

• X₃₈₄ = complement of X₆₆₅₅

• X₃₈₄ = anticomplement of X₆₆₅₆

• X₃₈₄ = antigonal image of X₃₇₈₈₈

• X₃₈₄ = circumcircle-inverse of X₃₇₈₉₆

• X₃₈₄ = orthocentroidal-circle-inverse of X₅₀₂₅

• X₃₈₄ = eigencenter of cevian triangle of X₆₉₄

• X₃₈₄ = eigencenter of anticevian triangle of X₃₈₅

• X₃₈₄ = antitomic image of X₁₆₁₀₁

• X₃₈₄ = polar conjugate of X₃₇₈₉₂

• X₃₈₄ = polar conjugate of X₃₇₈₉₂

• X₃₈₄ = antigonal conjugate of X₃₇₈₈₈

• X₃₈₄ = intersection of tangents at PU(1) to hyperbola {A, B, C,X₉₉,PU(1)}

• X₃₈₄ = perspector of ABC and symmedial triangle of 1st Brocard triangle

• X₃₈₄ = perspector of 1st and 3rd Brocard triangles

• X₃₈₄ = X₄₀₂₇-of-6th-Brocard-triangle

• X₃₈₄ = perspector of 1st Brocard triangle and cross-triangle of ABC and 1st Brocard triangle

• X₃₈₄ = perspector of 3rd Brocard triangle and cross-triangle of ABC and 3rd Brocard triangle

• X₃₈₄ = pole of Brocard axis wrt conic {X₁₃,X₁₄,X₁₅,X₁₆,X₇₆}}

• X₃₈₄ = homothetic center of 1st and 6th Brocard triangles

• X₃₈₄ = endo-homothetic center of 1st and 6th anti-Brocard triangles

• X₃₈₄ = X_i-anticomplementary conjugate of X_j for these (i,j): {1241, 21275}, {35567, 21305}

• X₃₈₄ = X_i-Ceva conjugate of X_j for these (i,j): {2, 37895}, {694, 385}, {14617, 3329}

• X₃₈₄ = X_i-cross conjugate of X_j for these (i,j): {4074, 9230}, {6657, 36432}

• X₃₈₄ = X₉₄₉₁-vertex conjugate of X₁₉₅₆₆

• X₃₈₄ = X₃₃₃-beth conjugate of X₂₄₆₁₀

• X₃₈₄ = X₁-zayin conjugate of X₆₉₅

• X₃₈₄ = X_i-isoconjugate of X_j for these (i,j): {1, 695}, {2, 9288}, {6, 9285}, {31, 9229}, {32, 9239}, {76, 9236}, {1966, 14946}

• X₃₈₄ = X_i-Hirst inverse of X_j for these (i,j): {2, 6660}, {4, 15145},{6, 19566},{1915, 16985},{1916, 9468}

• X₃₈₄ = cevapoint of X_i and X_j for these (i,j): {1, 8866}, {3, 3499}, {4, 8863}, {32, 3492}, {39, 3491}, {76, 8790}, {194, 2896}, {695, 8874}, {3493, 8871}, {3494, 8872}, {3496, 6196}, {3501, 8865}, {3503, 8862}, {8861, 8870}, {8864, 8873}, {8867, 8875}

• X₃₈₄ = crosspoint of X₈₃ and X₃₁₁₅

• X₃₈₄ = crosssum of X_i and X_j for these (i,j): {39, 3118}, {76, 24734}

• X₃₈₄ = crossdifference of every pair of points on line {647, 3221}

• X₃₈₄ = barycentric product X_i*X_j for these {i,j}: {1, 1965}, {6, 9230}, {31, 1925}, {75, 1582}, {76, 1915}, {83, 4074}, {305, 11380}, {561, 1932}, {1799, 12143}, {1916, 16985}, {6660, 16101}, {9229, 36432}, {19566, 22252}

• X₃₈₄ = barycentric quotient X_i/X_j for these {i,j}: {1, 9285}, {2, 9229}, {6, 695}, {31, 9288}, {75, 9239}, {560, 9236}, {1582, 1}, {1915, 6}, {1925, 561}, {1932, 31}, {1965, 75}, {3618, 3866}, {4074, 141}, {6660, 3505}, {9230, 76}, {9468, 14946}, {11380, 25}, {12143, 427}, {16985, 385}

• X₃₈₄ = {X₂,X₃}-harmonic conjugate of X₇₈₂₄

• X₃₈₄ = {X₂,X₄}-harmonic conjugate of X₅₀₂₅

• X₃₈₄ = {X₂,X₅}-harmonic conjugate of X₃₂₉₆₇

• X₃₈₄ = {X₂,X₂₀}-harmonic conjugate of X₇₇₉₁

• X₃₈₄ = {X_i,X_j}-harmonic conjugate of X_i for 1060 triples (i,j,k), where 2 <= i < j <= 37868 and 2 <= k <= 37868

Notes

As a point on the Euler line, X₃₈₄ has Shinagawa coefficients \((E + F)^{2} - S^{2}, 2S^{2})\).

Contributed by John Horton Conway, email, 1998.

Let A'B'C' be the 1st Brocard triangle. Let A″ be the cevapoint of B' and C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₃₈₄. (Randy Hutson, December 11, 2015)

Let L_a be the line through the 2nd Brocard circle intercepts (other than PU(1)) of lines AP(1) and AU(1); define L_b and L_c cyclically. Let A' = L_b∩L_c, B' = L_c∩L_a, C' = L_a∩L_b. The triangle A'B'C' is here introduced as the 6th Brocard triangle; A'B'C' is homothetic to the 1st Brocard triangle at X₃₈₄, homothetic to the 1st anti-Brocard triangle at X₃, and perspective to the 3rd Brocard triangle at X₃₈₄. The 2nd Brocard circle of ABC is the 1st Lemoine circle of A'B'C'. (Randy Hutson, December 26, 2015)

Let A'B'C' be the 1st Brocard-reflected triangle. Let A″ be the isogonal conjugate of A', and define B″ and C″ cyclically. The lines A'A″, B'B″, C'C″ concur in X₃₈₄. (Randy Hutson, June 7, 2019)