HYACINTHOS 28993

[Kadir Altintas]:, 

 

Let ABC be a triangle with centroid G centroid

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

The circle with diameter GD intersects the NPC of ABC again at  A'''; define B''',C''' cyclically

 

Prove that lines A''A''', B''B''', C''C''' concur at a point Q on Euler line of ABC.

 

Romantics of Geometry, problem 2972 b

 

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[Ercole Suppa]

 

Q = COMPLEMENT OF X(26255)

 

= a^6-4 a^4 b^2-a^2 b^4+4 b^6-4 a^4 c^2+18 a^2 b^2 c^2-4 b^4 c^2-a^2 c^4-4 b^2 c^4+4 c^6 : : (barys)

 

= (36 R^2-5 SW)S^2 -3 SB SC SW : : (barys)

=  As a point on the Euler line, X() has Shinagawa coefficients:  [4 e - 5 f, -3 e -3 f]

= lies on these lines: {2,3}, {6,13857}, {115,21448}, {125,599}, {126,3014}, {524,26869}, {542,6090}, {2393,5650}, {2549,24855}, {2790,23234}, {5024,9745}, {5642,14982}, {6054,9717}, {7998,14984}, {9140,15066}, {9155,9759}, {11064,11179}, {15059,21766}, {15080,20772}, {15131,19153}, {16187,25561}, {19136,22112}

= complement of X(26255)

 

= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {2,376,468}, {2,381,11284}, {2,858,381}, {2,10989,1995}, {2,16063,7426}, {381,3534,18325}, {427,468,6623}, {549,5159,2}, {1368,5159,18531}, {1995,10989,3830}, {7426,16063,3534}, {11284,21312,25}

 

= (6-9-13) search numbers [3.68795996897347955, 2.80871111455792854, -0.00596319846656198185]

 

 

 

Best regards

Ercole Suppa