HYACINTHOS 26315

[Tran Quang Hung]:

 

Let ABC be a triangle with Feuerbach points Fe, Fa, Fb, Fc.

 

The excircles (Ia), (Ib), (Ic) touch BC, CA, AB at A', B', C', resp.

 

The lines FaA',FbB',FcC' bound a triangle A''B''C''.

 

Then the NPC circle of the triangle A''B''C'' passes through the Feuerbach point Fe of ABC.

Center of the circle?

 

 

[Peter Moses]:

 

 

 

Hi Antreas,

 

a^8 b^2-2 a^7 b^3-2 a^6 b^4+6 a^5 b^5-6 a^3 b^7+2 a^2 b^8+2 a b^9-b^10-2 a^8 b c+3 a^6 b^3 c+a^5 b^4 c+2 a^4 b^5 c-2 a^3 b^6 c-5 a^2 b^7 c+a b^8 c+2 b^9 c+a^8 c^2-6 a^6 b^2 c^2+3 a^5 b^3 c^2+8 a^4 b^4 c^2+4 a^3 b^5 c^2-6 a^2 b^6 c^2-7 a b^7 c^2+3 b^8 c^2-2 a^7 c^3+3 a^6 b c^3+3 a^5 b^2 c^3+4 a^3 b^4 c^3+5 a^2 b^5 c^3-5 a b^6 c^3-8 b^7 c^3-2 a^6 c^4+a^5 b c^4+8 a^4 b^2 c^4+4 a^3 b^3 c^4+8 a^2 b^4 c^4+9 a b^5 c^4-2 b^6 c^4+6 a^5 c^5+2 a^4 b c^5+4 a^3 b^2 c^5+5 a^2 b^3 c^5+9 a b^4 c^5+12 b^5 c^5-2 a^3 b c^6-6 a^2 b^2 c^6-5 a b^3 c^6-2 b^4 c^6-6 a^3 c^7-5 a^2 b c^7-7 a b^2 c^7-8 b^3 c^7+2 a^2 c^8+a b c^8+3 b^2 c^8+2 a c^9+2 b c^9-c^10:: 

on the Euler line {2,3}

 

circle passes through X{11,5520}.

 

Best regards,

Peter Moses.

BC, A*B*C* are parallelogic.  Centers:  P(A->A*) = X(36)  &  P(A*->A) = X(4511)

 

César Lozada