Question 18
(1) $O$ is the circumcenter of the triangle $ABC$. The triangle is rotated about $O$ to give a new triangle $A'B'C'$. The lines $AB$ and $A'B'$ intersect at $C''$, $BC$ and $B'C'$ intersect at $A''$, and $CA$ and $C'A'$ intersect at $B''$. Show that $A''B''C''$ is similar to $ABC$.
(2) $O$ is the center of the circle through $ABCD$. $ABCD$ is rotated about $O$ to give the quadrilateral $A'B'C'D'$. Prove that the intersection points of corresponding sides form a parallelogram.
(2) $O$ is the center of the circle through $ABCD$. $ABCD$ is rotated about $O$ to give the quadrilateral $A'B'C'D'$. Prove that the intersection points of corresponding sides form a parallelogram.