RU-ALLRUSSIAN 1975 Problem 1

(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.