HYACINTHOS 23726

Dear friends,

 

I post the following, because ETC does not contain the point P.

 

Let ABC be a triangle and I its incenter.

 

Denote:

 

Ra, Rb, Rc = the reflection points of I in BC, CA, AB respectively.

 

D, E, F = the reflection points of I in Ra, Rb, Rc respectively.

 

Euler A, Euler B, Euler C = the Euler lines of DBC, ECA, FAB respectively.

 

These Euler lines concur in a point P.

 

1 st barycentric of P =

 

-7a^4+6a^3(b+c)+a^2(5b^2-13bc+5c^2)+a(-6b^3+5b^2c+5bc^2-6c^3)+2(b^2-c^2)^2.

 

Search number of triangle {6, 9, 13} = 0.723715568912377..

 

Addition:

 

The lines parallel to Euler A, Euler B, Euler C through A, B, C respectively concur in X(80).

 

Best regards, Seiichi Kirikami