#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
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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