Πέμπτη 31 Οκτωβρίου 2019

ADGEOM 4032 * ADGEOM 4033

#4032

Dear Mister Francisco Javier,

Dual of this problem:

 Let ABC be a triangle, construct on the sides of ABC three equilateral triangles BA'C, CB'A, AC'B either all outward or all inward. Three lines through A, B, C and parallel to B'C' , C'A', A'B' form a triangle A''B''C''. Then show that: The Euler lines of BA''C, CB''A, AC''B are concurrent. The angle of these Euler line equals to 60 degree. Which is the point of concurrence in ETC?

Best regards
Sincerely
Dao Thanh Oai
 
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#4033
 
These points are not in ETC.
 
I only find X1352 collinear with them.
 
Best regards, 
Francisco Javier.
 
 
Coordinates in the outwards case: 
 
{15 a^12 - 69 a^10 b^2 + 135 a^8 b^4 - 144 a^6 b^6 + 87 a^4 b^8 - 
  27 a^2 b^10 + 3 b^12 - 69 a^10 c^2 + 162 a^8 b^2 c^2 - 
  129 a^6 b^4 c^2 + 24 a^4 b^6 c^2 + 12 a^2 b^8 c^2 + 135 a^8 c^4 - 
  129 a^6 b^2 c^4 + 51 a^4 b^4 c^4 - 3 a^2 b^6 c^4 - 18 b^8 c^4 - 
  144 a^6 c^6 + 24 a^4 b^2 c^6 - 3 a^2 b^4 c^6 + 30 b^6 c^6 + 
  87 a^4 c^8 + 12 a^2 b^2 c^8 - 18 b^4 c^8 - 27 a^2 c^10 + 3 c^12 - 
  8 Sqrt[3] a^10 S + 18 Sqrt[3] a^8 b^2 S - 16 Sqrt[3] a^6 b^4 S + 
  8 Sqrt[3] a^4 b^6 S - 6 Sqrt[3] a^2 b^8 S + 4 Sqrt[3] b^10 S + 
  18 Sqrt[3] a^8 c^2 S - 4 Sqrt[3] a^6 b^2 c^2 S - 
  10 Sqrt[3] a^4 b^4 c^2 S + 2 Sqrt[3] a^2 b^6 c^2 S - 
  12 Sqrt[3] b^8 c^2 S - 16 Sqrt[3] a^6 c^4 S - 
  10 Sqrt[3] a^4 b^2 c^4 S + 20 Sqrt[3] a^2 b^4 c^4 S + 
  8 Sqrt[3] b^6 c^4 S + 8 Sqrt[3] a^4 c^6 S + 
  2 Sqrt[3] a^2 b^2 c^6 S + 8 Sqrt[3] b^4 c^6 S - 
  6 Sqrt[3] a^2 c^8 S - 12 Sqrt[3] b^2 c^8 S + 4 Sqrt[3] c^10 S, 
 3 a^12 - 27 a^10 b^2 + 87 a^8 b^4 - 144 a^6 b^6 + 135 a^4 b^8 - 
  69 a^2 b^10 + 15 b^12 + 12 a^8 b^2 c^2 + 24 a^6 b^4 c^2 - 
  129 a^4 b^6 c^2 + 162 a^2 b^8 c^2 - 69 b^10 c^2 - 18 a^8 c^4 - 
  3 a^6 b^2 c^4 + 51 a^4 b^4 c^4 - 129 a^2 b^6 c^4 + 135 b^8 c^4 + 
  30 a^6 c^6 - 3 a^4 b^2 c^6 + 24 a^2 b^4 c^6 - 144 b^6 c^6 - 
  18 a^4 c^8 + 12 a^2 b^2 c^8 + 87 b^4 c^8 - 27 b^2 c^10 + 3 c^12 + 
  4 Sqrt[3] a^10 S - 6 Sqrt[3] a^8 b^2 S + 8 Sqrt[3] a^6 b^4 S - 
  16 Sqrt[3] a^4 b^6 S + 18 Sqrt[3] a^2 b^8 S - 8 Sqrt[3] b^10 S - 
  12 Sqrt[3] a^8 c^2 S + 2 Sqrt[3] a^6 b^2 c^2 S - 
  10 Sqrt[3] a^4 b^4 c^2 S - 4 Sqrt[3] a^2 b^6 c^2 S + 
  18 Sqrt[3] b^8 c^2 S + 8 Sqrt[3] a^6 c^4 S + 
  20 Sqrt[3] a^4 b^2 c^4 S - 10 Sqrt[3] a^2 b^4 c^4 S - 
  16 Sqrt[3] b^6 c^4 S + 8 Sqrt[3] a^4 c^6 S + 
  2 Sqrt[3] a^2 b^2 c^6 S + 8 Sqrt[3] b^4 c^6 S - 
  12 Sqrt[3] a^2 c^8 S - 6 Sqrt[3] b^2 c^8 S + 4 Sqrt[3] c^10 S, 
 3 a^12 - 18 a^8 b^4 + 30 a^6 b^6 - 18 a^4 b^8 + 3 b^12 - 
  27 a^10 c^2 + 12 a^8 b^2 c^2 - 3 a^6 b^4 c^2 - 3 a^4 b^6 c^2 + 
  12 a^2 b^8 c^2 - 27 b^10 c^2 + 87 a^8 c^4 + 24 a^6 b^2 c^4 + 
  51 a^4 b^4 c^4 + 24 a^2 b^6 c^4 + 87 b^8 c^4 - 144 a^6 c^6 - 
  129 a^4 b^2 c^6 - 129 a^2 b^4 c^6 - 144 b^6 c^6 + 135 a^4 c^8 + 
  162 a^2 b^2 c^8 + 135 b^4 c^8 - 69 a^2 c^10 - 69 b^2 c^10 + 
  15 c^12 + 4 Sqrt[3] a^10 S - 12 Sqrt[3] a^8 b^2 S + 
  8 Sqrt[3] a^6 b^4 S + 8 Sqrt[3] a^4 b^6 S - 12 Sqrt[3] a^2 b^8 S + 
  4 Sqrt[3] b^10 S - 6 Sqrt[3] a^8 c^2 S + 2 Sqrt[3] a^6 b^2 c^2 S + 
  20 Sqrt[3] a^4 b^4 c^2 S + 2 Sqrt[3] a^2 b^6 c^2 S - 
  6 Sqrt[3] b^8 c^2 S + 8 Sqrt[3] a^6 c^4 S - 
  10 Sqrt[3] a^4 b^2 c^4 S - 10 Sqrt[3] a^2 b^4 c^4 S + 
  8 Sqrt[3] b^6 c^4 S - 16 Sqrt[3] a^4 c^6 S - 
  4 Sqrt[3] a^2 b^2 c^6 S - 16 Sqrt[3] b^4 c^6 S + 
  18 Sqrt[3] a^2 c^8 S + 18 Sqrt[3] b^2 c^8 S - 8 Sqrt[3] c^10 S}
 
Coordinates in the inwards case:
 
{15 a^12 - 69 a^10 b^2 + 135 a^8 b^4 - 144 a^6 b^6 + 87 a^4 b^8 - 
  27 a^2 b^10 + 3 b^12 - 69 a^10 c^2 + 162 a^8 b^2 c^2 - 
  129 a^6 b^4 c^2 + 24 a^4 b^6 c^2 + 12 a^2 b^8 c^2 + 135 a^8 c^4 - 
  129 a^6 b^2 c^4 + 51 a^4 b^4 c^4 - 3 a^2 b^6 c^4 - 18 b^8 c^4 - 
  144 a^6 c^6 + 24 a^4 b^2 c^6 - 3 a^2 b^4 c^6 + 30 b^6 c^6 + 
  87 a^4 c^8 + 12 a^2 b^2 c^8 - 18 b^4 c^8 - 27 a^2 c^10 + 3 c^12 + 
  8 Sqrt[3] a^10 S - 18 Sqrt[3] a^8 b^2 S + 16 Sqrt[3] a^6 b^4 S - 
  8 Sqrt[3] a^4 b^6 S + 6 Sqrt[3] a^2 b^8 S - 4 Sqrt[3] b^10 S - 
  18 Sqrt[3] a^8 c^2 S + 4 Sqrt[3] a^6 b^2 c^2 S + 
  10 Sqrt[3] a^4 b^4 c^2 S - 2 Sqrt[3] a^2 b^6 c^2 S + 
  12 Sqrt[3] b^8 c^2 S + 16 Sqrt[3] a^6 c^4 S + 
  10 Sqrt[3] a^4 b^2 c^4 S - 20 Sqrt[3] a^2 b^4 c^4 S - 
  8 Sqrt[3] b^6 c^4 S - 8 Sqrt[3] a^4 c^6 S - 
  2 Sqrt[3] a^2 b^2 c^6 S - 8 Sqrt[3] b^4 c^6 S + 
  6 Sqrt[3] a^2 c^8 S + 12 Sqrt[3] b^2 c^8 S - 4 Sqrt[3] c^10 S, 
 3 a^12 - 27 a^10 b^2 + 87 a^8 b^4 - 144 a^6 b^6 + 135 a^4 b^8 - 
  69 a^2 b^10 + 15 b^12 + 12 a^8 b^2 c^2 + 24 a^6 b^4 c^2 - 
  129 a^4 b^6 c^2 + 162 a^2 b^8 c^2 - 69 b^10 c^2 - 18 a^8 c^4 - 
  3 a^6 b^2 c^4 + 51 a^4 b^4 c^4 - 129 a^2 b^6 c^4 + 135 b^8 c^4 + 
  30 a^6 c^6 - 3 a^4 b^2 c^6 + 24 a^2 b^4 c^6 - 144 b^6 c^6 - 
  18 a^4 c^8 + 12 a^2 b^2 c^8 + 87 b^4 c^8 - 27 b^2 c^10 + 3 c^12 - 
  4 Sqrt[3] a^10 S + 6 Sqrt[3] a^8 b^2 S - 8 Sqrt[3] a^6 b^4 S + 
  16 Sqrt[3] a^4 b^6 S - 18 Sqrt[3] a^2 b^8 S + 8 Sqrt[3] b^10 S + 
  12 Sqrt[3] a^8 c^2 S - 2 Sqrt[3] a^6 b^2 c^2 S + 
  10 Sqrt[3] a^4 b^4 c^2 S + 4 Sqrt[3] a^2 b^6 c^2 S - 
  18 Sqrt[3] b^8 c^2 S - 8 Sqrt[3] a^6 c^4 S - 
  20 Sqrt[3] a^4 b^2 c^4 S + 10 Sqrt[3] a^2 b^4 c^4 S + 
  16 Sqrt[3] b^6 c^4 S - 8 Sqrt[3] a^4 c^6 S - 
  2 Sqrt[3] a^2 b^2 c^6 S - 8 Sqrt[3] b^4 c^6 S + 
  12 Sqrt[3] a^2 c^8 S + 6 Sqrt[3] b^2 c^8 S - 4 Sqrt[3] c^10 S, 
 3 a^12 - 18 a^8 b^4 + 30 a^6 b^6 - 18 a^4 b^8 + 3 b^12 - 
  27 a^10 c^2 + 12 a^8 b^2 c^2 - 3 a^6 b^4 c^2 - 3 a^4 b^6 c^2 + 
  12 a^2 b^8 c^2 - 27 b^10 c^2 + 87 a^8 c^4 + 24 a^6 b^2 c^4 + 
  51 a^4 b^4 c^4 + 24 a^2 b^6 c^4 + 87 b^8 c^4 - 144 a^6 c^6 - 
  129 a^4 b^2 c^6 - 129 a^2 b^4 c^6 - 144 b^6 c^6 + 135 a^4 c^8 + 
  162 a^2 b^2 c^8 + 135 b^4 c^8 - 69 a^2 c^10 - 69 b^2 c^10 + 
  15 c^12 - 4 Sqrt[3] a^10 S + 12 Sqrt[3] a^8 b^2 S - 
  8 Sqrt[3] a^6 b^4 S - 8 Sqrt[3] a^4 b^6 S + 12 Sqrt[3] a^2 b^8 S - 
  4 Sqrt[3] b^10 S + 6 Sqrt[3] a^8 c^2 S - 2 Sqrt[3] a^6 b^2 c^2 S - 
  20 Sqrt[3] a^4 b^4 c^2 S - 2 Sqrt[3] a^2 b^6 c^2 S + 
  6 Sqrt[3] b^8 c^2 S - 8 Sqrt[3] a^6 c^4 S + 
  10 Sqrt[3] a^4 b^2 c^4 S + 10 Sqrt[3] a^2 b^4 c^4 S - 
  8 Sqrt[3] b^6 c^4 S + 16 Sqrt[3] a^4 c^6 S + 
  4 Sqrt[3] a^2 b^2 c^6 S + 16 Sqrt[3] b^4 c^6 S - 
  18 Sqrt[3] a^2 c^8 S - 18 Sqrt[3] b^2 c^8 S + 8 Sqrt[3] c^10 S}
 
 
 Francisco Javier Garcia Capitan
 

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