The present work is aimed at mathematicians and scientists, as well as enthusiasts of geometry. No special expertise is required. More people than commonly believed yearn for an activity that can provide assurance that the human being has a share in a purely intellectual world. Independent mathematical thinking, particularly geometrizing, can become such an activity if pursued vigorously and then examined with all cognitive faculties. Some elementary concepts are developed that seem important and particularly suitable for thoroughly and engagingly familiarizing a wider circle of mathematics enthusiasts with this field. The presentations are written with the conviction that thinking within projective geometry is valuable.