as core mathematics

This method for finding the centre, and by extension the equation, of a circle given three non-colinear points, brings the ancient textbook master Euclid onto Descartes’ coordinate plane and right  into the 21 century classroom. 

Geometric Progressions, essential for Core AS mathematicians, appear in Victorian textbook, Synopsis of Elementary Results.  In the case where the sum to infinity exists, there is a nice proof by similar triangles.