Question 1
Trapezium $ABCD$ with a longer base $AB$ is inscribed in the circle $k$. Let $A_0$, $B_0$ be respectively the midpoints of segments $BC$, $CA$. Let $N$ be the foot of the altitude from the point $C$ to $AB$, and $G$ the centroid of the triangle $ABC$. Circle $k_1$ goes through $A_0$ and $B_0$ and touches the circle $k$ in the point $X$, different than $C$. Prove that the points $D$, $G$, $N$ and $X$ are collinear.