SHARYGIN 2005 Problem 5

There are two parallel lines $p_1$ and $p_2$ . Points $A$ and $B$ lie on $p_1$ , and $C$ on $p_2$ . We will move the segment $BC$ parallel to itself and consider all the triangles $AB'C '$ thus obtained. Find the locus of the points in these triangles: a) points of intersection of heights, b) the intersection points of the medians, c) the centers of the circumscribed circles.