HYACINTHOS 28990

 

[Kadir Altintas]: Romantics of Geometry, problem 2972

 

Let ABC be a triangle with centroid G.

Let DEF be the medial triangle of ABC

AD intersects the NPC of ABC at A'. Define B', C' cyclically

The circle with diameter GA' intersects the NPC of ABC again at  A''. Define B'',C'' cyclically

 

Prove:

1. Medial triangle of ABC is perspective with A''B''C''. Perspector P lies on Euler line of ABC

2. Orthic triangle of ABC is perspective with A''B''C''. Perspector Q lies on Euler line of ABC

 

Romantics of Geometry, problem 2972

 

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[Ercole Suppa]

 

1. 

 

P = COMPLEMENT OF X(1995)

 

= a^4 b^2-b^6+a^4 c^2-8 a^2 b^2 c^2+b^4 c^2+b^2 c^4-c^6 : : (barys)

 

= (12 R^2 - SW)S^2 - SB SC SW : : (barys)

 

= lies on these lines: {2,3}, {9,26933}, {11,4319}, {12,4320}, {57,21015}, {69,26869}, {95,8901}, {120,29857}, {122,216}, {125,126}, {182,11064}, {230,5063}, {323,1353}, {325,11059}, {343,3819}, {373,5480}, {394,11245}, {495,5297}, {496,7292}, {574,24855}, {597,13857}, {599,5486}, {612,15888}, {908,25365}, {1177,15131}, {1184,5319}, {1194,9607}, {1196,7765}, {1503,5651}, {1506,15820}, {1899,15069}, {2549,16317}, {3003,3815}, {3054,9722}, {3266,3933}, {3291,5254}, {3292,8550}, {3564,15066}, {3580,7998}, {3589,22112}, {3763,5646}, {3818,16187}, {3917,13567}, {3925,23304},  {5092,5972}, {5640,21850}, {5888,15059}, {5891,16003}, {5913,9465}, {5971,14929}, {6090,6776}, {6292,15667}, {6509,26905}, {7085,20266}, {7668,11168}, {7999,26879}, {8262,16789}, {8758,29639}, {9680,18289}, {9777,18928}, {9820,13336}, {10192,22352}, {10293,10706}, {11061,15106}, {11513,13884}, {11514,13937}, {14569,15466}, {15082,24206}, {15135,22151}, {16511,19510}, {17818,26937}, {19924,20192}, {25406,26864}, {26866,27509}

 

= midpoint of X(i) and X(j) for these {i,j}: {1995,16063}, {15066,18911}

 

= reflection of X(i) in X(j) for these {i,j}: {10301,1995}, {16063,10300}

= complement of X(1995)

 

= complementary conjugate of X(8542)

 

= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {2,3,468}, {2,4,11284}, {2,22,6677}, {2,858,5}, {2,1368,427}, {2,1370,5020}, {2,5189,16042}, {2,7386,25}, {2,7396,7392}, {2,7484,7499}, {2,7485,6676}, {2,7496,7495}, {2,8889,7539}, {2,10300,10301}, {2,16051,5094}, {2,16063,1995}, {3,6816,1885}, {5,20,1906}, {5,858,427}, {5,1368,858}, {25,7386,7667}, {125,5650,141}, {140,5159,2}, {140,11585,7399}, {140,12362,17928}, {427,468,235}, {468,1885,25}, {632,13371,7405}, {1368,5159,11585}, {1370,5020,428}, {1594,14789,5}, {5067,15559,5}, {5092,5972,13394}, {5169,7493,11799}, {6676,7734,7485}, {6677,10691,22}, {7392,7396,5064}, {7495,7496,549}, {8889,15809,427}

 

= (6-9-13) search numbers [3.47293609061144866, 2.59425447461629871, 0.241748573044734459]

 

 

2. 

 

Q = X(5094)

 

 

Best regards

Ercole Suppa