Question 22
Let points $A_1$, $A_2$, and $A_3$ lie on the circle $\Gamma$ in counter-clockwise order, and let $P$ be a point in the same plane. For $i \in \{1, 2, 3\}$, let $\tau_i$ denote the counter-clockwise rotation of the plane centred at $A_i$, where the angle of the rotation is equal to the angle at vertex $A_i$ in $\triangle A_1A_2A_3$. Further, define $P_i$ to be the point $\tau_{i+2}(\tau_i(\tau_{i+1}(P)))$, where indices are taken modulo 3 (i.e., $\tau_4 = \tau_1$ and $\tau_5 = \tau_2$).
Prove that the radius of the circumcircle of $\triangle P_1P_2P_3$ is at most the radius of $\Gamma$.
Prove that the radius of the circumcircle of $\triangle P_1P_2P_3$ is at most the radius of $\Gamma$.