Angle at the Center & Power of a Point

This lecture covers the concept of the angle at the center and the power of a point relative to a circle, demonstrating the theorems of Pascal and Brianchon in three cases. It explains Euclid's demonstration that the products of PA.PB & PC.PD on two arbitrary transversals are equal, when P is inside the circle, outside the circle, and when none of the transversals are tangent to the circle. It also explores the construction of the radical axis and the perpendicularity of the line of equal power points to the line joining the circle centers. The lecture concludes by discussing the generalization of the radical axis from Euclid's Elements and De Architectura. Various geometric configurations and demonstrations are presented.