Σάββατο 9 Μαρτίου 2024

A PROOF OF MORLEY THEOREM

Thanasis Gakopoulos - Debabrata Nag, Morley Theorem ̶ PLAGIOGONAL Approach of Proof

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.

Gakopoulos - Nag

Παρασκευή 19 Ιανουαρίου 2024

X(61637), X(61638)

X(61637) = ISOGONAL CONJUGATE Χ(61638)

Barycentrics   

See 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)

Barycentrics   

See 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

ΒΙΒΛΙΑ

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

Mail Antreas P. Hatzipolakis

A PROOF OF MORLEY THEOREM

Thanasis Gakopoulos - Debabrata Nag, Morley Theorem ̶ PLAGIOGONAL Approach of Proof Abstract: In this work, an attempt has been made b...