THREE CURIOUS MIDPOINTS

Let ABC be a triangle, P a point and A'B'C' the cevian triangle of P Ma = the midpoint of AbAc. Similarly Mb, Mc
Which is the locus of P such that ABC, MaMbMc are orthologic?
Antreas Hatzipolakis

Study:
Francisco Javier García Capitán: Three curious midpoints
FJGC

BILOGIC TRIANGLES

Antreas Hatzipolakis, Bilogic Triangles by Construction
Euclid 6242

Study by Francisco Javier García Capitán: Bilogic triangles by construction

Cosmology of Plane Geometry: Concepts and Theorems

Alexander Skutin,Tran Quang Hung, Antreas Hatzipolakis, Kadir Altintas: Cosmology of Plane Geometry: Concepts and Theorems>

ΨΗΦ. Cosmology of Plane Geometry: Concepts and Theorems
ΨΗΦ. Cosmology of Plane Geometry: Concepts and Theorems

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ΨΗΦ. Cosmology of Plane Geometry: Concepts and Theorems
ΨΗΦ. Cosmology of Plane Geometry: Concepts and Theorems

Fermat-Torricelli and Napoleon Points

Thanasis Gakopoulos, Debabrata Nag: Fermat-Torricelli and Napoleon Points of a Triangle
A PLAGIOGONAL Approach

ΨΗΦ Fermat-Torricelli and Napoleon Points

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

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)

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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)

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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}

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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}

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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)}}

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