HYACINTHOS 28858

[Tran Quang Hung]:

 
Let ABC be a triangle.

Euler line meets BC, CA, AB at A', B', C', resp.

Perpendiculars from A', B', C' to BC, CA, AB resp. bound a triangle A"B"C".

Then X(1552) of A"B"C" lies on the Euler line of ABC.

Which is this point?


[Angel Montesdeoca]:
   
    ***  W = MIDPOINT OF X(107) AND X(1304)
   
  X(1304) is the  center of similitude of the triangles ABC and A"B"C".
   
     W = (a^2-b^2) (a^2-c^2) (2 a^14-2 a^12 (b^2+c^2)+9 a^8 (b^2-c^2)^2 (b^2+c^2)-8 a^4 (b^2-c^2)^4 (b^2+c^2)+(b^2-c^2)^6 (b^2+c^2)+a^10 (-7 b^4+16 b^2 c^2-7 c^4)+4 a^6 (b^2-c^2)^2 (b^4-5 b^2 c^2+c^4)+a^2 (b^2-c^2)^4 (b^4+8 b^2 c^2+c^4)) :  :

 

= lies on these lines: {2,3}, {107,523}, {110,8057}, {112,9209}, {250,3233}, {476,1301}, {935,9064}, {1289,9060}, {1302,10423}, {1552,2777}, {2693,23239}, {6070,17986}, {6530,16319}, {6587,23964}, {6716,16177}, {10420,30249}, {16166,20626}
 
= midpoint of X(107) and X(1304)

= reflection of X(i) in X(j), for these {i, j}: {1552,18809}, {16177,6716}, {16386,27089}

 (6 - 9 - 13) - search numbers  of W: (0.396216444365828, -0.474348707516369, 3.78619061278841).
 
 Angel Montesdeoca