Question 12
Let $A$, $B$, $C$, $A'$, $B'$, $C'$ be distinct points on the plane satisfying $ABC \cong A'B'C'$ and the point $G$ be the centroid of the triangle $ABC$. If the circle of center $A'$ passing through $G$ and the circle of diameter $[AA']$ intersect at point $A_1$, the circle of center $B'$ passing through $G$ and the circle of diameter $[BB']$ intersect at point $B_1$, the circle of center $C'$ passing through $G$ and the circle of diameter $[CC']$ intersect at point $C_1$, show that
$$
AA_1^2 + BB_1^2 + CC_1^2 \le AB^2 + BC^2 + CA^2.
$$
$$
AA_1^2 + BB_1^2 + CC_1^2 \le AB^2 + BC^2 + CA^2.
$$