Two circles \(c\) and \(d\) are tangent at point \(B\). Two straight lines intersecting the first circle at points \(D\) and \(E\) and the second circle at points \(G\) and \(F\) are drawn through the point \(B\). Prove that \(ED\) is parallel to \(FG\).
Let \(ABCD\) be a square and let \(P\) be any point in the plane. For each side of the square, take its midpoint. Reflect \(P\) about each of these four midpoints. Show that the four reflected points form the vertices of a square.
The points \(A\) and \(B\) and the line \(l\) are given on a plane. On which trajectory does the intersection point of the medians of the triangles \(ABC\) move, if the point \(C\) moves along the line \(l\)?