NL-NWO 2007 Problem 1

Consider the equilateral triangle $ABC$ with $|BC| = |CA| = |AB| = 1$ . On the extension of side $BC$ , we define points $A_1$ (on the same side as B) and $A_2$ (on the same side as C) such that $|A_1B| = |BC| = |CA_2| = 1$ . Similarly, we define $B_1$ and $B_2$ on the extension of side $CA$ such that $|B_1C| = |CA| =|AB_2| = 1$ , and $C_1$ and $C_2$ on the extension of side $AB$ such that $|C_1A| = |AB| = |BC_2| = 1$ . Now the circumcentre of 4ABC is also the centre of the circle that passes through the points $A_1,B_2,C_1,A_2,B_1$ and $C_2$ . Calculate the radius of the circle through $A_1,B_2,C_1,A_2,B_1$ and $C_2$ .