MORLEY RELATED TRIANGLE CENTERS

MORLEY RELATED TRIANGLE CENTERS
BY ANTREAS HATZIPOLAKIS and CHRIS VAN TIENHOVEN, PETER MOSES, CESAR LOZADA

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Antreas Hatzipolakis and Chris van Tienhoven

X(6120) =  1st HATZIPOLAKIS-VAN TIENHOVEN POINT
X(6121) =  2nd HATZIPOLAKIS-VAN TIENHOVEN POINT
X(6122) =  3rd HATZIPOLAKIS-VAN TIENHOVEN POINT
X(6123) =  4th HATZIPOLAKIS-VAN TIENHOVEN POINT
X(6124) =  5th HATZIPOLAKIS-VAN TIENHOVEN POINT
X(6125) =  6th HATZIPOLAKIS-VAN TIENHOVEN POINT
X(66087) = HATZIPOLAKIS-VAN TIENHOVEN EQUILATERAL TRIANGLE CENTER
X(66089) = PERSPECTOR OF THESE TRIANGLES: 1st MORLEY AND HATZIPOLAKIS-VAN TIENHOVEN EQUILATERAL TRIANGLE

Antreas Hatzipolakis and Peter Moses

X(65155) =  X(2)X(15857)∩X(357)X(8065)
X(65156) =  HATZIPOLAKIS-MOSES EQUILATERAL TRIANGLE CENTER
X(65157) =  X(2)X(3276)∩X(5)X(3280)
X(65158) =  X(357)X(5456)∩X(358)X(1136)
X(65377) = CENTER OF HATZIPOLAKIS-MOSES-MORLEY HYPERBOLA
X(65378) = PERSPECTOR OF HATZIPOLAKIS-MOSES-MORLEY HYPERBOLA
X(65379) = X(357)X(1136)∩X(10632)X(16871)
X(65380) = ANTICOMPLEMENT OF X(65377)
X(65381) = 98TH HATZIPOLAKIS-MOSES-EULER POINT
X(65382) = TRILINEAR POLE OF LINE {6, 65378}
X(66329) = 1st HATZIPOLAKIS-MOSES TRISECTOR TRIANGLES PERSPECTOR
X(66330) = 2nd HATZIPOLAKIS-MOSES TRISECTOR TRIANGLES PERSPECTOR
X(66331) = 3rd HATZIPOLAKIS-MOSES TRISECTOR TRIANGLES PERSPECTOR
X(66332) = CENTER OF THE 1st MORLEY CONIC
X(66333) = PERSPECTOR OF THE 1st MORLEY CONIC
X(66365) = X(356)-CEVA CONJUGATE OF X(357)
X(66366) = X(3276)-CEVA CONJUGATE OF X(1136)
X(66367) = X(3277)-CEVA CONJUGATE OF X(1134)
X(66619) = X(3275)X(3602)∩X(3276)X(3606)

Antreas Hatzipolakis and César Lozada

X(66478) = CENTER OF THE 1st EULER-ROUSSEL EQUILATERAL TRIANGLE
X(66479) = CENTER OF THE 2nd EULER-ROUSSEL EQUILATERAL TRIANGLE
X(66480) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st EULER-ROUSSEL AND 1st MORLEY
X(66481) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd EULER-ROUSSEL AND 1st MORLEY

ETC

X(66630) = 1st EULER-LINE-DE-LONGCHAMPS-CIRCLE INTERSECTION

Barycentrics    (SB SC - a^2 SA) Sqrt[9 R^2 - 2 SW] - (2 SB SC - a^2 SA) Sqrt[4 R^2 - SW] : :

See Ivan Pavlov, euclid 7685.

X(66630) lies on the De Longchamps circle, the circumconic {{A, B, C, X(30786), X(34168)}} and this line: {2, 3}

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X(66631) = 2nd EULER-LINE-DE-LONGCHAMPS-CIRCLE INTERSECTION

Barycentrics    (SB SC - a^2 SA) Sqrt[9 R^2 - 2 SW] + (2 SB SC - a^2 SA) Sqrt[4 R^2 - SW] : :

See Ivan Pavlov, euclid 7685.

X(66631) lies on the De Longchamps circle, the circumconic {{A, B, C, X(30786), X(34168)}} and this line: {2, 3}

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JOHN H. CONWAY Papers

Bibliography of John H. Conway.

ARTICLES AND PAPERS

M.R. Boothroyd and John H. Conway, Problems Drive, 1959. Eureka 22 (October 1959) 15-17. (Solutions: 22-23).
EUREKA

John H. Conway and M.J.T. Guy, Π in four 4’s. Eureka 25 (October 1962) 18-19.
EUREKA

John H. Conway, Tomorrow is the Day After Doomsday. Eureka 36 (October 1973) 28-31.
EUREKA

John H. Conway, The Hunting of J4. Eureka 41 (Summer 1981), 46-54.
EUREKA

John H. Conway, The Weird and Wonderful Chemistry of Audioactive Decay. Eureka 46 (January 1986) 5-18.
EUREKA

John H. Conway, Forewod. In: G. Polya, How to solve it. A new aspect of mathematical method. Expanded version of the 1988 edition, with a new foreword by John H. Conway. Princeton Science Library. Princeton University Press, Princeton, NJ, 2004, p. xix-xxiv.
POLYA

John Conway, The Power of Mathematics. In: Alan Blackwell and David MacKay (Editors), Power (Darwin College Lectures, Series Number 16). Cambridge University Press 2006, pp 36-50
John Conway

Conway J, Ryba A. The Steiner-Lehmus angle-bisector theorem. The Mathematical Gazette. 2014;98(542):193-203. doi:10.1017/S0025557200001236
Conway-Ryba-Lehmus

Buchanan, A.G., Conway, J.H. (2017). An Island Tale for Young Anthropologists. In: Fitting, M., Rayman, B. (eds) Raymond Smullyan on Self Reference. Outstanding Contributions to Logic, vol 14. Springer, Cham. https://doi.org/10.1007/978-3-319-68732-2_10
Buchanan-Conway

Douglas Hofstadter, FOREWORD

Douglas Hofstadter, FOREWORD
In: Clark Kimberling, Triangle Centers and Central Triangles. Congressus Numerantum, vol. 129, August, 1998. Winnipeg, Canada

Douglas Hofstadter

REGULAR POLYGONS AND EULER LINES

Let A1A2A3 be an equilateral triangle and Pa point.
Denote:
1, 2, 3 = the Euler lines of PA1A2,PA2A3, PA3A1, resp.
1,2,3 are concurrent.
Hyacinthos 21592

GENERALIZATION
Let A1A2A3...A3k be a regular 3k-gon and P a point.
Denote:
1, 2, 3,.... 3k = the Euler lines of PA1A2, PA2A3, PA3A4,.....PA3kA1, resp.
These triads are concurrent
1, k+1, 2k+1
2, k+2, 2k+2
3, k+3, 2k+3
4, k+4, 2k+4
......
k, 2k, 3k

k = 2 (Regular Hexagon)
1, 3, 5
2, 4, 6

Mail Antreas P. Hatzipolakis

X(5159) updated

X(5159) = INVERSE-IN-NINE-POINT-CIRCLE OF X(1368)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b² + c² - a²)(3b⁴ + 3c⁴ - 2a⁴ + a²b² + a²c² - 6b²c²)
Barycentrics    (a^2-b^2-c^2)*(2*a^4-3*(b^2-c^2)^2-a^2*(b^2+c^2)) : :
Barycentrics    SA*(6*SB*SC-SB*SW-SC*SW) : :
X(5159) = 9*X(2)-X(23), 3*X(2)+X(858), 5*X(2)-X(7426), 7*X(2)+X(10989), 3*X(2)+5*X(30745), 6*X(2)-X(37897), 7*X(2)-X(37904), 11*X(2)-3*X(37907), 9*X(2)+X(46517), 5*X(2)+X(47311), 11*X(2)-X(47312), 6*X(2)+X(47315), 9*X(2)-2*X(47316), 3*X(2)+2*X(47629), X(3)+3*X(2072), X(3)-3*X(10257), 2*X(3)-3*X(16976), 3*X(3)+X(18323), 5*X(3)+3*X(18403), 7*X(3)-3*X(44246)

As a point on the Euler line, X(5159) has Shinagawa coefficients (E -5*F, -E - F)

X(5159) = (inverse-in-de-Longchamps-circle of X(22)) = (radical trace of nine-point circle and de-Longchamps circle) = (radical trace of polar circle and de-Longchamps circle) = (reflection of X(23) in de Longchamps line)    (Randy Hutson, August-September, 2013)

For a construction of X(5159), see David Nguyen and Ercole Suppa, euclid 7508.

X(5159) lies on these lines: {1, 47489}, {2, 3}, {6, 47546}, {8, 47533}, {10, 47492}, {39, 47182}, {69, 40920}, {98, 44877}, {114, 16177}, {122, 16188}, {125, 3292}, {126, 38971}, {127, 31655}, {131, 46436}, {141, 8542}, {193, 47463}, {216, 3055}, {230, 3284}, {265, 55981}, {325, 16315}, {339,3266}, {343, 8538}, {373, 37511}, {385, 47237}, {394, 8548}, {511, 6723}, {523, 4885}, {524, 6698}, {525, 22264}, {551, 47491}, {575, 23292}, {576, 13567}, {577, 3054}, {597, 47460}, {599, 47280}, {625,40544}, {647, 47256}, {850, 47248}, {892, 16103}, {895, 15128}, {1007, 2452}, {1038, 7286}, {1040, 5160}, {1092, 61544}, {1125, 51725}, {1154, 64689}, {1351, 37643}, {1352, 59767}, {1353, 26869}, {1503, 5972}, {1514, 36518}, {1531, 38727}, {1560, 52951}, {1568, 38729} ,{1698, 47321}, {1853, 59543}, {1899, 38398}, {1992, 47462}, {2697, 44060}, {2770, 58096}, {3066, 38136}, {3167, 23291}, {3231, 14965}, {3241, 47536}, {3258, 31842}, {3291, 14961}, {3576, 47469}, {3580, 15059}, {3589, 20113}, {3618, 32220}, {3619, 47279}, {3620, 47281}, {3634, 37613}, {3679, 47490}, {3763, 32113}, {3815, 5158}, {3818, 61507}, {4669, 47564}, {5093, 63081}, {5099, 31275}, {5203, 53895}, {5272, 10149}, {5297, 18447}, {5622, 41615}, {5650, 9967}, {5651, 18358}, {5886, 47471}, {6000, 65095}, {6390, 15398}, {6425, 18289}, {6426, 18290}, {6509, 15850}, {6593, 15116}, {6688, 58481}, {6697, 13562}, {6721, 62490}, {6776, 62708}, {7292, 18455}, {7736, 47184}, {7745, 15820}, {7763, 59766}, {7868, 16325}, {7925, 47155}, {7998, 18438}, {8263, 23327}, {8280, 43879}, {8281, 43880}, {8705, 11574}, {8780, 32064}, {8854, 53513}, {8855, 53516}, {9019, 35370}, {9140, 32272}, {9148, 47159}, {9165, 44401}, {9300, 15860}, {9306, 23332}, {9716, 45968}, {9745, 63633}, {9771, 16333}, {9775, 46637}, {9820, 18914}, {9970, 15131}, {10116, 15129},{10173, 11594}, {10192, 64196}, {10256, 46987}, {10415, 34897}, {10510, 47558}, {10516, 47474}, {10564, 23515}, {10634, 43103}, {10635, 43102}, {11059, 41009}, {11245, 11422}, {11427, 53092}, {11433, 11482}, {11477, 26958}, {11513, 32789}, {11514, 32790}, {11580, 22121}, {12041, 58885}, {12079, 14919}, {12358, 40685}, {12900, 14915}, {13202, 20725}, {13292, 15120}, {13445, 15029}, {13561, 31831}, {13611, 65620}, {13754, 16270}, {13857, 41586}, {13869, 30741}, {14156, 15115}, {14341, 41357}, {14389, 51732}, {14561, 47571}, {14984, 60774}, {15025, 50435}, {15027, 22115}, {15034, 25739}, {15066, 61545}, {15124, 50708}, {15125, 15311}, {15139, 15142}, {15448, 29012}, {15471, 32300}, {15491, 16324}, {15526, 22110}, {15533, 47466}, {15812, 47355}, {15819, 47568}, {15905, 62992}, {16092, 39061}, {16227, 64854}, {16303, 31489}, {16310, 44529}, {16313, 50666}, {16760, 34844}, {18583, 37648}, {18911, 61690}, {19126, 32217}, {19131, 22112}, {19862, 51693}, {19875, 47488}, {19883, 47495}, {20190, 58447}, {20208, 37690}, {20299, 59659}, {20582, 47446}, {21243, 53415}, {21356, 47551}, {21358, 47276}, {21639, 53778}, {21850, 61506}, {21968, 64059}, {21970, 51212}, {22104, 34841}, {22151, 32251}, {23878, 47247}, {25055, 47472}, {29181, 32223}, {29639, 47178}, {31174, 46983}, {31274, 47326}, {31277, 47004}, {31279, 47173}, {31804, 64181}, {31843, 46664}, {32111, 64101}, {32227, 34153}, {32269, 51360}, {32411, 63709}, {32767, 64035}, {33522, 55602}, {33924, 47164}, {34128, 51391}, {34988, 44533}, {35259, 39884}, {35707, 58437}, {35968, 42426}, {36900, 47257}, {37638,48876}, {37644, 61624}, {37647, 62698}, {37688, 41008}, {37689, 38292}, {38028, 47476}, {38047, 47506}, {38053, 47507}, {38057, 47508}, {38098, 47534}, {38110, 54012}, {38314, 47493}, {38317, 47581}, {38795, 51425}, {39220, 52058}, {39899, 64177}, {40112, 41724}, {40341, 47465}, {40349, 40350}, {40995, 63098}, {41139, 46998}, {42424, 53832}, {44381, 47239}, {44436, 47213}, {46264, 61680}, {46686, 58871}, {46982, 63440}, {46986, 47245}, {46989, 47254}, {47169, 62196}, {47175, 47259}, {47207, 65760}, {47249, 47261}, {47253, 47442}, {47271 ,54320}, {47352, 47458}, {47447, 63121}, {47451, 51128}, {47456, 52238}, {47461, 51171}, {47494, 53620}, {47499, 47761}, {47500, 47760}, {47537, 51071}, {47541, 59373}, {47544, 48310}, {47575, 59403}, {47576, 59404}, {51358, 59661}, {52520, 63632}, {52987, 61646}, {53777, 62375}, {58451, 58639}, {59399, 63084}, {60414, 60415}, {60420, 60424}, {60421, 60425}, {60422, 60426}, {60423, 60427}

X(5159) = midpoint of X(i) and X(j) for these (i, j): {2, 47097}, {3, 10297}, {5, 15122}, {23, 46517}, {69, 47277}, {125, 11064}, {325, 16315}, {403, 47090}, {427, 16387}, {441, 37987}, {468, 858}, {625, 40544}, {850, 47248}, {892, 16103}, {1529, 46620}, {2071, 10151}, {2072, 10257}, {3153, 37931}, {3679, 47593}, {5189, 37899}, {5196, 47098}, {5203, 53895}, {6390, 51258}, {7426, 47311}, {7575, 47341}, {9148, 47159}, {10295, 47339}, {10510, 47558}, {10989, 37904}, {12041, 58885}, {13202, 20725}, {13473, 16386}, {13857, 44569}, {15115, 15123}, {15118, 19510}, {16188, 47570}, {18323, 47308}, {18572, 47335}, {22110, 46980}, {23323, 34152}, {31174, 46983}, {32269, 51360}, {36170, 56370}, {37897, 47315}, {37900, 47095}, {37911, 47629}, {37938, 44452}, {37950, 47336}, {38971, 54075}, {46686, 58871}, {46982, 63440}, {47091, 47096}, {47092, 62344}, {47280, 47552}, {47310, 54995}, {47312, 47314}, {47612, 47613}
X(5159) = reflection of X(i) in X(j) for these (i, j): (4, 63821), (23, 47316), (403, 44912), (468, 37911), (858, 47629), (10257, 63860), (15471, 32300), (16976, 10257), (32217, 47454), (37897, 468), (37899, 47630), (37910, 37897), (37935, 44452), (37942, 44911), (37984, 5), (41357, 14341), (47004, 47262), (47114, 16976), (47239, 44381), (47261, 47249), (47296, 6723), (47315, 858), (47338, 11799), (47442, 47253), (47457, 3589), (47549, 47460), (51725, 1125), (65154, 6677)
X(5159) = complement of X(468)
X(5159) = inverse-in-nine-point circle of X(1368)
X(5159) = inverse-in-{circumcircle, nine-point circle}-inverter of X(20)
X(5159) = inverse-in-complement-of-polar-circle of X(2)
X(5159) = intersection, other than A, B, C, of circumconics {A, B, C, X(3), X(40349)} and {A, B, C, X(4), X(53419)}
X(5159) = barycentric product X(i)*X(j) for these (i,j): (69, 53419), (76, 21639), (264, 40349), (305, 40350), (525, 53351), (5485, 53778)
X(5159) = barycentric quotient X(i)/X(j) for these {i,j}: {40349, 3}, {40350, 25}, {53351, 648}, {53419, 4}, {53778, 1992}
X(5159) = trilinear product X(i)*X(j) for these (i,j): (63, 53419), (75, 21639), (92, 40349), (304, 40350), (656, 53351), (53778, 55923)
X(5159) = trilinear quotient X(i)/X(j) for these (i,j): (21639, 31), (40349, 48), (40350, 1973), (53351, 162), (53419, 19), (53778, 36277)
X(5159) = (X(i), X(j))-harmonic conjugate of X(k) for these (i, j, k): (2, 20, 52290), (2, 22, 52297), (2, 427, 6677), (2, 468, 37911), (23, 468, 47316), (597, 47549, 47460), (599, 47280, 47552), (3291, 47298, 43291), (3618, 32220, 47459), (5651, 45303, 18358), (10510, 62376, 47558), (24855, 47298, 3291), (26869, 37645, 1353), (30786, 37804, 62310), (37648, 61743, 18583), (37804, 62310, 6390), (51360, 61691, 32269), (59767, 61735, 1352)

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PYTHAGOREAN THEOREM

From a New Criterion of a Cyclic Quadrilateral (NCCQ) to Pythagorean System –– An Alternative Study
Athanasios Gakopoulos, Farsala, Greece
Debabrata Nag, Kolkata, India

Gak_Nag_Pythagoras_07-11_2024