Let ABC be a triangle and r1,r2,r3 three not equal line segments.
a1 = the circle centered at A with radius r1. Similarly ....
a1b2 = the radical axis of the circles a1 and b2. Similarly .....
Six Radical centers:
(a1,b2,c3), (a1,b3,c2), (a2,b3,c1), (a2,b1,c3), (a3,b1,c2), (a3,b2,c1)
Six other points of concurrent radical axes:
(a1b2,b3c1,c2a3), (a1b3,b2c1,c3a2), (a2b3,b1c2,c3a1), (a2b1,b3c2,c1a3), (a3b1,b2c3,c1a2),(a3b2,b1c3,c2a1)
The 12gon has opposite sides parallel and equal. It is inscribed on a conic centered at the circumcenter O = radical center of (a1,b1,c1) and (a2,b2,c2) and (a3,b3,c3)
Antreas P. Hatzipolakis. 29 Jan. 2013