Explore the concept of the core circle, a crucial variant of the unit circle with a diameter spanning the unit interval [0,1]. Delve into its significance in understanding the projective line and its role as a bridge between projective and Euclidean planar geometry. Learn about basic facts and formulas related to the core circle, including how quadrances between points on it align with projective quadrances between corresponding projective 1-points. Discover a formula for calculating the quadrea of a triangle on the core circle. Review the unit circle, examine the advantages of the core circle, and study its parametrization. Investigate three points on the core circle, prove the Core circle theorem, and explore the Core circle quadrance theorem in this comprehensive 39-minute lecture on rational geometry.