HYACINTHOS 27485

 

[Antreas P. Hatzipolakis]:

Let ABC be a triangle.

Denote:

(Na), (Nb), (Nc) = the NPCs of NBC, NCA, NAB, resp.

(Nba), (Nca) = the reflections of (Nb), (Nc) in ANa, resp.

A' = the radical center of (Na), (Nba), (Nca)

[they concur]

Similarly B', C'.

The points A', B', C' and the Poncelet point of ABCN are concyclic.

Center of the circle?

 
[César Lozada]:

 

Center: X(1487)

 

The antipode in this circle of X(137) (= Poncelet point of ABCN) is:

 

Z = reflection of X(137) in X(1487)

= (28*cos(2*A)-15*cos(4*A)-17*co s(6*A)-12*cos(8*A)+31/2)*cos( B-C)+8*(3*cos(A)-cos(3*A)-cos( 5*A)-2*cos(7*A))*cos(2*(B-C))+ (-3*cos(2*A)+10*cos(4*A)-15* cos(6*A)+43/2)*cos(3*(B-C))+4* (2*cos(A)+cos(3*A)-cos(5*A))* cos(4*(B-C))+(7*cos(2*A)-3*cos (4*A)+1)*cos(5*(B-C))-16*cos( 5*A)-10*cos(7*A)-2*cos(9*A)+ 10*cos(A) : : (trilinears)

= on lines: {137, 1487}, {3530, 12026}

= reflection of X(137) in X(1487)

= [ 2.9188377951483710, 2.6972753251690160, 0.4261641205681195 ]

 

No other ETC centers found on this circle.

 

César Lozada