Good work on this entire series of articles on Euclidean geometry. While cartesian geometry offers more ease at handling many more complex problems, the sheer charm and intuition which Euclidean geometry offers makes me return to it constantly . Do check my article which uses Euclidean concepts for an intuition related to polygons and circles https://medium.com/intuition/apothem-is-to-a-polygon-what-radius-is-to-a-circle-8adaf290c54c