angle at center is double angle subtended at circumference

You can clearly see a circle in red color which passes through the point B and has its center at A. Points C and D are two other points present on the circumference of this circle. Let us observe the angle subtended by the arc CD on the center and any point on the circumference. Angle CBD (α) is the angle made by the arc CD with any point (B) present on the circumference. Angle CAD (β) is the angle made by the arc CD with center of the circle A. You can move all the five free points A, B, C, D and E. Move the points A, B, C and D and see how the size and position of the circle gets changed.