In a question on Mathematics Stack Exchange, Moguzya asked for tangents to a moving circle. My answer finds up to four of them, as shown below.

The large red point controls , the initial position of the circle. Its radius can be changed by dragging the rim of the circle around that red point. The smaller red point controls and , the speed and direction of the movement of the circle. More precisely, the position of that point is the center of the circle at . The central circle can be changed (drag to change radius) to control , the speed along the tangent.

The solutions are drawn in red for (point of contact in the past) and dark green for . The parameters and are printed for each solution.