Abstract: In this work, an attempt has been made by the authors to present a PLAGIOGONAL
approach to prove the Morley Theorem involving the intersecting trisectors of the angles of a
scalene triangle. The objective of the present work is to also establish the non-orthogonal coordinates
of the vertices of Morley triangle.
ΜΑΘΗΜΑΤΙΚΑ
PERSONAL MATHEMATICS NOTEBOOK
Σάββατο 9 Μαρτίου 2024
A PROOF OF MORLEY THEOREM
Παρασκευή 19 Ιανουαρίου 2024
X(61637), X(61638)
X(61637) = ISOGONAL CONJUGATE Χ(61638)
BarycentricsSee Floor van Lamoen and Francisco, euclid 6085.
X(61637) lies on this line:
X(61637) = isogonal comjugate of X(61638)
X(61638) = X(2)X(3)∩X(195)X(15109)
BarycentricsSee Floor van Lamoen and Francisco, euclid 6085.
X(61638) lies on these lines:
X(61638) = isogonal comjugate of X(61637)
Κυριακή 24 Δεκεμβρίου 2023
X(61298) - X(61300)
X(61298) = X(5)X(39494)∩X(1116)X(10224)
Barycentrics (b-c)*(b+c)*(a^2*b^2*(a^2-b^2)^4*(a^2+b^2)+(a^2-b^2)^2*(a^8-3*a^2*b^6-b^8)*c^2+(-3*a^10+a^8*b^2+6*a^6*b^4-4*a^4*b^6+3*b^10)*c^4+(2*a^8-3*a^6*b^2-4*a^4*b^4-2*b^8)*c^6+(2*a^6+5*a^4*b^2-2*b^6)*c^8-(3*a^4+a^2*b^2-3*b^4)*c^10+(a-b)*(a+b)*c^12) : :See Antreas Hatzipolakis and Ivan Pavlov, euclid 6029.
X(61298) lies on these lines: {5, 39494}, {1116, 10224}, {1594, 39512}, {10280, 39503}, {11615, 39509}, {18308, 50136}, {32478, 33332}
X(61299) = X(26)X(1853)∩X(30)X(511)
Barycentrics 2*a^10+a^6*(b^2-c^2)^2-4*a^8*(b^2+c^2)-(b^2-c^2)^4*(b^2+c^2)+a^2*(b^2-c^2)^2*(b^4+3*b^2*c^2+c^4)+a^4*(b^6+2*b^4*c^2+2*b^2*c^4+c^6) : :See Antreas Hatzipolakis and Ivan Pavlov, euclid 6029.
X(61299) lies on these lines: {4, 13353}, {5, 22352}, {22, 34514}, {23, 15027}, {26, 1853}, {30, 511}, {52, 45732}, {125, 37936}, {140, 13419}, {143, 7553}, {146, 46445}, {154, 31181}, {156, 11206}, {186, 38728}, {265, 37925}, {382, 7592}, {428, 13364}, {546, 44829}, {548, 45286}, {1495, 37938}, {1533, 44283}, {1658, 23329}, {2937, 34826}, {3530, 17712}, {3627, 11750}, {3853, 15807}, {5073, 12174}, {5189, 22115}, {5498, 46265}, {5876, 16659}, {5899, 13171}, {5946, 7540}, {6723, 44900}, {6756, 12006}, {7502, 11550}, {7514, 36990}, {7555, 21243}, {7574, 14157}, {7575, 38729}, {7728, 46440}, {7748, 39524}, {10096, 32237}, {10113, 47096}, {10116, 14449}, {10192, 13371}, {10193, 15331}, {10263, 11264}, {10540, 20125}, {10610, 15559}, {10627, 12134}, {11455, 18564}, {11565, 12241}, {11695, 13163}, {11818, 46264}, {11819, 13630}, {12046, 23411}, {12107, 20299}, {12121, 37944}, {12140, 37931}, {12168, 35452}, {12278, 17800}, {12362, 45958}, {12605, 32137}, {13292, 16982}, {13363, 13490}, {13421, 32358}, {13451, 43573}, {13565, 34002}, {13598, 45970}, {13851, 43893}, {14791, 31383}, {14927, 18420}, {15061, 37940}, {15088, 37942}, {15761, 23324}, {16621, 52073}, {16655, 45959}, {16881, 18128}, {17714, 18381}, {18282, 32767}, {18403, 51548}, {18572, 51403}, {19154, 23327}, {20379, 47342}, {20396, 37897}, {21849, 45969}, {21969, 45730}, {22251, 51393}, {23325, 44278}, {23328, 48368}, {23332, 44213}, {23335, 32171}, {31305, 32140}, {33533, 46448}, {35018, 44862}, {37924, 50435}, {40111, 51360}, {45186, 45731}, {45971, 46850}, {47341, 51425}, {52397, 54042}
X(61299) = pole of line {125, 15026} with respect to the Jerabek hyperbola
X(61299) = pole of line {110, 7525} with respect to the Stammler hyperbola
X(61299) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {30, 1503, 1154}, {10263, 34224, 11264}, {10540, 46450, 51391}, {11264, 34224, 45734}, {29012, 44407, 30}
X(61300) = X(51)X(476)∩X(511)X(930)
Barycentrics a^2*(a^2*b^2*(a^2-b^2)^4-2*a^2*b^2*(a^2-b^2)^2*(a^2+b^2)*c^2+(a^8+2*a^6*b^2+2*a^2*b^6+b^8)*c^4-(a^2+b^2)*(3*a^4+a^2*b^2+3*b^4)*c^6+(3*a^4+4*a^2*b^2+3*b^4)*c^8-(a^2+b^2)*c^10)*(a^10*c^2-b^4*c^2*(b^2-c^2)^3+a^8*(b^4-2*b^2*c^2-4*c^4)+a^6*(-3*b^6+2*b^4*c^2+2*b^2*c^4+6*c^6)+a^4*(3*b^8-4*b^6*c^2+2*b^2*c^6-4*c^8)-a^2*(b-c)*(b+c)*(b^8-3*b^6*c^2+b^4*c^4-b^2*c^6+c^8)) : :See Antreas Hatzipolakis and Ivan Pavlov, euclid 6029.
X(61300) lies on the circumcircle and these lines: {51, 476}, {98, 1510}, {99, 1154}, {511, 930}, {512, 1141}, {567, 691}, {933, 34397}, {1291, 5012}, {2715, 2965}, {22456, 32002}, {46966, 54034}
X(61300) = intersection, other than A, B, C, of circumconics, {{A, B, C, X(51), X(512)}}, {{A, B, C, X(74), X(98)}}, {{A, B, C, X(187), X(567)}}, {{A, B, C, X(249), X(288)}}, {{A, B, C, X(511), X(1510)}}, {{A, B, C, X(1157), X(5012)}}, {{A, B, C, X(2065), X(57639)}}, {{A, B, C, X(14587), X(50946)}} and {{A, B, C, X(51480), X(52179)}}
X(61139)
X(61139) = X(4)X(54)∩X(24)X(125)
Barycentrics 2*a^10-4*a^8*(b^2+c^2)+a^4*(b^2-c^2)^2*(b^2+c^2)-(b^2-c^2)^4*(b^2+c^2)+a^6*(b^2+c^2)^2+a^2*(b^4-c^4)^2 : :X(61139) = -3*X[2]+2*X[44829], -3*X[51]+2*X[6146], -2*X[389]+3*X[7576], -3*X[428]+2*X[12241], -3*X[568]+2*X[10116], -3*X[3060]+2*X[10112], -3*X[3830]+2*X[12897], -2*X[5446]+3*X[7540], -4*X[5480]+5*X[52789], -9*X[5946]+8*X[50476], -X[6241]+3*X[18559], -3*X[9730]+4*X[31830], -4*X[10110]+3*X[12022], -5*X[10574]+X[40241], -3*X[11245]+4*X[11745], -2*X[11565]+3*X[13364], -2*X[12605]+3*X[15030], -4*X[13348]+3*X[52397], -2*X[13474]+3*X[16658], -2*X[13488]+3*X[16654], -2*X[13598]+3*X[34603], -2*X[13630]+3*X[38322], -3*X[16194]+2*X[52070], -4*X[16625]+3*X[45968], -4*X[18128]+5*X[37481], -3*X[38321]+2*X[40647], -4*X[43588]+3*X[45730], -4*X[44870]+3*X[52069]
See Antreas Hatzipolakis and Ivan Pavlov, euclid 6016.
X(61139) lies on these lines: {2, 44829}, {3, 2918}, {4, 54}, {5, 1495}, {20, 1352}, {24, 125}, {26, 18474}, {30, 5562}, {32, 51363}, {51, 6146}, {52, 11819}, {64, 67}, {68, 41586}, {74, 52102}, {113, 18377}, {115, 52436}, {143, 45731}, {154, 7507}, {155, 382}, {156, 44288}, {159, 1593}, {182, 7544}, {185, 1503}, {186, 20299}, {235, 13851}, {265, 18378}, {378, 34785}, {389, 7576}, {403, 18383}, {427, 13367}, {428, 12241}, {511, 14516}, {539, 6243}, {542, 5889}, {568, 10116}, {569, 11818}, {1092, 14790}, {1141, 11816}, {1147, 31723}, {1181, 18494}, {1204, 13399}, {1209, 7502}, {1370, 43652}, {1498, 12173}, {1514, 3853}, {1568, 10539}, {1594, 10282}, {1598, 18396}, {1658, 34514}, {1853, 3515}, {1885, 16621}, {1899, 7487}, {2070, 5449}, {2777, 12281}, {2937, 6288}, {2980, 22261}, {3060, 10112}, {3146, 12278}, {3331, 7747}, {3357, 35471}, {3410, 7691}, {3426, 17800}, {3518, 25739}, {3542, 44082}, {3547, 35268}, {3564, 14531}, {3581, 52104}, {3627, 30522}, {3818, 7503}, {3830, 12897}, {5064, 11425}, {5094, 17821}, {5446, 7540}, {5448, 10540}, {5480, 52789}, {5576, 18475}, {5651, 6643}, {5899, 48675}, {5907, 12225}, {5944, 39504}, {5946, 50476}, {6000, 6240}, {6143, 10182}, {6241, 18559}, {6247, 21663}, {6293, 9973}, {6696, 37931}, {6746, 41589}, {6815, 46264}, {6995, 18945}, {7391, 13346}, {7399, 22352}, {7401, 43650}, {7488, 21243}, {7505, 23325}, {7512, 41171}, {7517, 9927}, {7545, 43821}, {7553, 44665}, {7574, 18350}, {7575, 13561}, {7577, 26882}, {7684, 45256}, {7685, 45257}, {7687, 18394}, {7715, 44106}, {7731, 13423}, {8779, 27376}, {9306, 37444}, {9714, 14852}, {9730, 31830}, {9908, 12293}, {10018, 32767}, {10110, 12022}, {10117, 32357}, {10193, 17506}, {10263, 13417}, {10301, 15873}, {10312, 15340}, {10574, 40241}, {10594, 18390}, {10605, 34780}, {10610, 50138}, {10984, 18420}, {10996, 14927}, {11202, 37119}, {11204, 35503}, {11245, 11745}, {11403, 45015}, {11430, 15559}, {11432, 34564}, {11438, 11457}, {11441, 52842}, {11442, 31304}, {11449, 31074}, {11464, 52295}, {11563, 18379}, {11565, 13364}, {11645, 38323}, {12084, 16163}, {12106, 43817}, {12107, 34826}, {12295, 44271}, {12429, 33586}, {12605, 15030}, {13348, 52397}, {13366, 31804}, {13371, 51393}, {13434, 19130}, {13474, 16658}, {13488, 16654}, {13491, 45971}, {13598, 34603}, {13630, 38322}, {14049, 19504}, {14118, 41482}, {14585, 27371}, {15019, 43838}, {15122, 43898}, {15750, 40686}, {15811, 44438}, {16194, 52070}, {16195, 37638}, {16252, 23047}, {16625, 45968}, {17701, 23315}, {18128, 37481}, {18376, 35488}, {18404, 46261}, {18405, 37197}, {18488, 18570}, {18563, 45118}, {18907, 56866}, {19124, 36989}, {19137, 41257}, {19558, 39604}, {20987, 51756}, {21844, 25563}, {22802, 35480}, {22804, 46029}, {23208, 54003}, {23294, 44673}, {23329, 32534}, {23335, 51394}, {24206, 37126}, {26917, 47485}, {26937, 32064}, {26958, 55578}, {29323, 54040}, {31726, 52863}, {32345, 37954}, {34417, 37122}, {34609, 35602}, {34776, 39588}, {37198, 48905}, {37452, 43586}, {38321, 40647}, {38791, 57271}, {43588, 45730}, {43907, 47335}, {44831, 46728}, {44870, 52069}, {51434, 51509}
X(61139) = midpoint of X(i) and X(j) for these {i,j}: {12290, 34797}, {3146, 12278}, {6240, 16659}
X(61086) = reflection of X(i) in X(j) for these {i,j}: {125, 12140}, {185, 3575}, {1885, 16621}, {11381, 16655}, {11750, 5}, {12225, 5907}, {12289, 13403}, {13491, 45971}, {18560, 13474}, {21659, 4}, {3, 45286}, {3574, 32332}, {34224, 389}, {34799, 10112}, {4, 13419}, {44076, 5446}, {45186, 7553}, {45731, 143}, {52, 11819}, {5562, 12134}, {6146, 6756}
X(61139) = anticomplement of X(44829)
X(61086) = X(i)-Dao conjugate of X(j) for these {i, j}: {44829, 44829}
X(61139) = pole of line {23286, 44808} with respect to the circumcircle
X(61139) = pole of line {389, 427} with respect to the Jerabek hyperbola
X(61139) = pole of line {3049, 12077} with respect to the orthic inconic
X(61139) = pole of line {1614, 5562} with respect to the Stammler hyperbola
X(61139) = pole of line {7750, 46724} with respect to the Wallace hyperbola
X(61139) = intersection, other than A, B, C, of circumconics {{A, B, C, X(67), X(38808)}}, {{A, B, C, X(1614), X(5562)}} and {{A, B, C, X(6662), X(8884)}}
X(61139) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 12254, 15033}, {4, 12289, 13403}, {4, 1614, 18388}, {4, 19467, 11424}, {4, 26883, 51403}, {4, 31383, 26883}, {4, 6759, 43831}, {4, 8884, 6747}, {4, 9833, 184}, {30, 12134, 5562}, {30, 16655, 11381}, {235, 41362, 13851}, {1092, 14790, 51360}, {1204, 14216, 13399}, {1495, 11572, 5}, {1503, 3575, 185}, {1885, 16621, 32062}, {3060, 34799, 10112}, {6146, 6756, 51}, {6240, 16659, 6000}, {7540, 44076, 5446}, {7553, 44665, 45186}, {10539, 18569, 1568}, {10540, 31724, 5448}, {11442, 31304, 46730}, {12289, 13403, 21659}, {12290, 34797, 2777}, {13289, 44795, 125}, {13403, 18400, 12289}, {13419, 18400, 4}, {14216, 18533, 1204}, {16658, 18560, 13474}, {17845, 36990, 1593}, {18388, 45185, 1614}, {18394, 44958, 7687}, {18400, 32332, 3574}, {37122, 39571, 34417}, {44407, 45286, 3}
Δευτέρα 18 Δεκεμβρίου 2023
X(61083), X(61084)
X(61083) = ISOGONAL CONJUGATE OF X(61084)
Barycentrics (SB + SC)*(SA*SB - S*Sqrt[SA*SB])*(SA*SC - S*Sqrt[SA*SC]) : :See Costas Vittas, Antreas Hatzipolakis and Peter Moses, euclid 6066.
X(61083) lies on the cubic K006, the curves Q039 and Q117 and this line: {4, 61084}
X(61083) = isogonal conjugate of X(61084)
X(61084) = ISOGONAL CONJUGATE OF X(61083)
Barycentrics (SB + SC)*(SA*SB + S*Sqrt[SA*SB])*(SA*SC + S*Sqrt[SA*SC]) : :See Costas Vittas, Antreas Hatzipolakis and Peter Moses, euclid 6066.
X(61084) lies on the cubic K006, the curves Q039 and Q117 and this line: {4, 61083}
X(61084) = isogonal conjugate of X(61083)
Κυριακή 15 Οκτωβρίου 2023
Let ABC be a triangle, D a point on BC and r, r_b, r_c the inradii of ABC, ABD, ACD, resp. 1. AB _|_ BC ( ie A = 90 d.), AD _|_ BC (ie AD is the a-altitude) ==> r^2 = r_b^2 + r_c^2
(Old theorem)
2. ΨΗΦ. Πρόβλημα 19
Πέμπτη 7 Σεπτεμβρίου 2023
ΣΤΥΛ. Γ. ΡΗΓΟΠΟΥΛΟΣ
Στυλιανού Γ. Ρηγοπούλου, Στοιχεία Γεωμετρίας διά τους μαθητάς των Γυμνασίων. Εν Αθήναις, Εκδότης: Δ. Χ. Τερζόπουλος 1910.
ΕΒΕ
Στυλιανού Γ. Ρηγοπούλου, Στοιχεία Γεωμετρίας διά τους μαθητάς των Γυμνασίων. Εν Αθήναις, Βιβλιοπωλείον Ιωάννου Ν. Σιδέρη 1914.
Στυλιανού Γ. Ρηγοπούλου, Στοιχειώδης Άλγεβρα προς χρήσιν των Γυμνασίων. Εν Αθήναις, Εκδότης: Δ. Χ. Τερζόπουλος 1914.
ΔΙΑΘΗΚΗ
ΨΗΦ. Ρηγόπουλος_ Διαθήκη
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