A hidden circle in the paths of a family of projectiles

It is already known that an ellipse passes through the apexes of a family of the parabolic paths of projectiles shot from a point with constant speed but different angles of projection. In this article we describe a method to show that a circle passes through the focii of such a family of projectile paths whose center is the point of projection. In this method, we don’t use vectors, or calculus or Newton’s laws of motion.