Explore the projective parametrization of the unit circle and its applications in geometry in this 45-minute mathematics lecture. Delve into the extension of rational parametrization, deriving formulas for areas of triangles and quadrilaterals circumscribed by the unit circle. Learn about the projective parametrization using complex numbers, geometry on the unit circle, and the quadrea of triangles. Discover the unit circle quadrilateral quadrea theorem and the cyclic quadrilateral quadrea theorem. Gain insights into circle geometry and its coherent analysis in the context of other conics.