Coincidence detection of two curves or two surfaces has wide application in computer-aided design (CAD) and computer-aided geometric design (CAGD). Proper reparameterization is the most complicated part in the detection. This paper presents and proves the efficient and necessary coincidence condition for two rational Bézier curves in a new way. It also proposes an effective and efficient proper reparameterization method, Algorithm 1, for detecting a rational Bézier curve which can be degenerated into a new one of a lower degree. A numerical proper reparameterization method, Algorithm 2, and examples are also presented. Algorithm 1 is up to ten times faster than other prevailing methods, and Algorithm 2 is twice as fast and half as close as other prevailing methods. New CAD systems using Algorithm 1 and Algorithm 2 will hold accuracy and little computation time.