HYACINTHOS 28317

[Tran Quang Hung]:

 

Let ABC be a triangle.

O is the Circumcenter.

O* is the isotomic conjugate of O.

A'B'C' is the cevian triangle of O.

A''B''C'' is the antipedal triangle of O wrt A'B'C'.

O'' is the circumcenter of triangle A''B''C''.

Prove that O'' lies on line OO*.

Please see AoPS

 

[Angel Montesdeoca]:

-----   O* =  X(264) is isotomic conjugate of O; lies on line X(3)X(95).
-----   O'' is circumcenter of triangle A''B''C''.
  
   O'' = a^2 (-(b^2-c^2)^2+a^2 (b^2+c^2)) (a^16-5 a^14 (b^2+c^2)+b^2 c^2 (b^2-c^2)^4 (b^4+4 b^2 c^2+c^4)+3 a^12 (3 b^4+7 b^2 c^2+3 c^4)-a^10 (5 b^6+33 b^4 c^2+33 b^2 c^4+5 c^6)+a^8 (-5 b^8+23 b^6 c^2+38 b^4 c^4+23 b^2 c^6-5 c^8)-a^4 (b^2-c^2)^2 (5 b^8+7 b^6 c^2+8 b^4 c^4+7 b^2 c^6+5 c^8)+a^2 (b^2-c^2)^2 (b^10-b^8 c^2-4 b^6 c^4-4 b^4 c^6-b^2 c^8+c^10)+a^6 (9 b^10-7 b^8 c^2-14 b^6 c^4-14 b^4 c^6-7 b^2 c^8+9 c^10)) :  :
  
= lies on these lines:  {3,95}, {1154,10606}, {7395,12012},  {11197,17928}

(6 - 9 - 13) - search numbers  of O'': (-36.4490708229672, 34.3019910430569, -3.28421970653172).
 
 Angel Montesdeoca