HYACINTHOS 25698

[Tran Quang Hung]:


Let ABC be a triangle with circumcenter O.

Ka,Kb,Kc are symmedian points of triangles OBC,OCA,OAB.

 

O' is isogonal conjugate of O wrt KaKbKc.

 

Then O' lies on Euler line of ABC.

 

Which is this point ?

 

 

[Angel Montesdeoca]:

Dear Tran Quang Hung,

O' is isogonal conjugate of O wrt KaKbKc lies on the lines :
{2,3}, {141,9925}, {156,5085}, {394,1493}, {569,5650}, {1173,2979}, {1350,10095}, {6101,10601}.

Barycentric coordinates:

(a^2 (a^8-2 a^6 (b^2+c^2)-8 a^4 b^2 c^2+2 a^2 (b^6+8 b^4 c^2+8 b^2 c^4+c^6)-(b^2-c^2)^2 (b^4+8 b^2 c^2+c^4)) : ... : ...),

with (6-9-13)-search number (3.25622880646691,2.3781188697 5912,0.491399661397189).

Best regards,
Angel Montesdeoca