Statement

Everyone knows that in Bugland rooms are an infinite plane surface in which each point is described by two coordinates (x, y): in particular, two bugs A and B are now waiting at coordinates (X_a, Y_a) and (X_b, Y_b). It is well known that in Bugland beds are perfect circles, described by the coordinates of their center and their radius: in particular, in the mentioned room there is a single bed with center C = (X_c, Y_c) and radius R. A and B want to have a secret meeting to plan the invasion of the room: thus, they need to move as silent as possible, according to any trajectory, and gather this way to a common meeting point (X_m, Y_m) (not necessarily of integer coordinates). Walking on the floor one unit makes 1 NU (Noise Unit): for example, walking from (1, 1) to (3, 1) makes 2 NU, and walking from (1, 1) to (3, 2) makes √5 NU. On the other hand, walking on the bed is completely silent. Help the two bugs A and B meet, using a path allowing the sum of their NUs to be minimal.