Explore the concept of the core circle, a crucial variant of the unit circle with a diameter spanning the unit interval [0,1], in this 33-minute lecture on rational geometry. Delve into the fundamental principles and formulas associated with this geometric object, examining its significance in bridging projective and Euclidean planar geometry. Discover how quadrances between points on the core circle align with projective quadrances between corresponding projective 1-points, and uncover an elegant formula for calculating the quadrea of a triangle on the core circle. Investigate topics such as relativistic velocity addition, Einstein's special theory of relativity, the distinction between velocities, relativistic rotation, and one-dimensional projective geometry to gain a comprehensive understanding of these interconnected mathematical concepts.