For tangency type problems, the actual shape of the rigid body undergoing rotational motion can be any closed curve. The traditional method of selecting a moving point based on the characteristic that the distance between the center and the contact point remains constant is insufficient for solving these problems. In order to address this type of problem, the instantaneous kinematic analysis method is utilized in teaching, which involves finding a special moving point. This point has an instantaneous relative velocity of zero, allowing us to obtain the required kinematic quantities easily and intuitively. This paper first proves the existence and uniqueness of this special moving point, then provides a method for determining this point, and finally presents three examples of solving such problems. This method has been applied in the teaching of theoretical mechanics at our university and has been well received by students due to its easy grasp and understanding.