HYACINTHOS 26388

[Tran Quang Hung] = buratinogigle (*)

 

Let ABC be a triangle, A'B'C' the pedal triangle of I and A"B"C" the cevian triangle of the Nagel point Na.

The Euler lines of the triangles NaA'A", NaB'B", NaC'C" are concurrent.

Point of concurrence?

(*) Here

 

 

[Angel Montesdeoca]:

 The Euler lines of the triangles NaA'A", NaB'B", NaC'C" are concurrent at 

W = ( a^7
         +5 a^6 (b+c)
         -a^5 (53 b^2-22 b c+53 c^2)
         -a^4 (29 b^3-129 b^2 c-129 b c^2+29 c^3)
         -a^3 (-79 b^4+64 b^3 c+222 b^2 c^2+64 b c^3-79 c^4)
         +a^2 (19 b^5-133 b^4 c+146 b^3 c^2+146 b^2 c^3-133 b c^4+19 c^5)
          -3 a (b^2-c^2)^2 (9 b^2-14 b c+9 c^2)
         +(b-c)^2 (b+c)^3 (5 b^2-6 b c+5 c^2) : ... : ...).

W =  2r(13r-6R) X(40) - 3(4r(r+3R)-s^2) X(376)

W = X(40)X(376) /\ X(3857)X(5530)

(6 - 9 - 13) - search numbers  of W: (8.21514432820893, 9.85564181395732, -6.97407723308250).

Angel Montesdeoca