Question 121
A convex hexagon $A_1B_1A_2B_2A_3B_3$ is inscribed in a circle $\Omega$ of radius $R$. The diagonals $A_1B_2$, $A_2B_3$, and $A_3B_1$ concur at $X$. For $i = 1, 2, 3$, let $\omega_i$ be the circle tangent to the segments $XA_i$ and $XB_i$, and to the arc $A_iB_i$ of $\Omega$ not containing other vertices of the hexagon; let $r_i$ be the radius of $\omega_i$.
a) Prove that $R \ge r_1 + r_2 + r_3$.
b) If $R = r_1 + r_2 + r_3$, prove that the six points where the circles $\omega_i$ touch the diagonals $A_1B_2$, $A_2B_3$, $A_3B_1$ are concyclic.
a) Prove that $R \ge r_1 + r_2 + r_3$.
b) If $R = r_1 + r_2 + r_3$, prove that the six points where the circles $\omega_i$ touch the diagonals $A_1B_2$, $A_2B_3$, $A_3B_1$ are concyclic.