Problems

Two circles with centres \(A\) and \(C\) are tangent at the point \(B\). The segment \(DE\) passes through the point \(B\). Prove that the tangent lines passing through the points \(D\) and \(E\) are parallel.

Let \(ABCD\) be a square, a point \(I\) a random point on the plane. Consider the four points, symmetric to \(I\) with respect to the midpoints of \(AB, BC, CD, AD\). Prove that these new four points are vertices of a square.