ADGEOM 5195 * ADGEOM 5196 * ADGEOM 5197

#5195

 

Let ABC be a triangle.

 

I see that X(1), X(5), X(186), and X(1785) are concyclic.

 

Is this circle known? Which is its center?

 

Best regards

Tran Quang Hung.

 

--------------------------------------------------

 

#5196

 

Dear Hung, 

 

the circle also goes through X11700 and X23961.

The first coordinate of the center is

a (a - b - c) (b - c) (a^5 + a^4 b - 2 a^3 b^2 - 2 a^2 b^3 + a b^4 +  b^5 + a^4 c + 2 a^2 b^2 c - 3 b^4 c - 2 a^3 c^2 + 2 a^2 b c^2 - a b^2 c^2 + 2 b^3 c^2 - 2 a^2 c^3 + 2 b^2 c^3 + a c^4 - 3 b c^4 +  c^5).

 

Best regards, 

Francisco Javier Garcia Capitan

 

--------------------------------------------------

 

#5197

 

 Dear Tran Quang Hung the center of circle is

 

W=a (b+c-a) (b-c) (a^5+a^4 b-2 a^3 b^2-2 a^2 b^3+a b^4+b^5+a^4 c+2 a^2 b^2 c-3 b^4 c-2 a^3 c^2+2 a^2 b c^2-a b^2 c^2+2 b^3 c^2-2 a^2 c^3+2 b^2 c^3+a c^4-3 b c^4+c^5) :: (barys)

 

= on lines X(i)X(j) for these {i,j}: {522,905},{900,5901},{3738,14315}

 

=  ETC search numbers: {2.67157859398621388, 0.505268931305329665, 2.05782663993320600}

 

Best regards 

Ercole Suppa