The fruit of the meditations of a young Lieutenant of the Engineers: A Theorem of Major Impact and Its Applications. Two Centuries from the Discovery of the Nine-Point Circle

The Nine-Point Circle Theorem was first obtained in a joint work written by C.-J. Brianchon and J.-V. Poncelet. Since the authors of this Theorem presented it in their original work as a principle in geometry, we are invited to reflect whether there are some direct applications of this Theorem in advanced Euclidean geometry. Our exploration only confirms the original authors’ foretelling intuition, since un nouveau principe à la géométrie élémentaire is expected to admit applications. We regard the Nine-Point Circle Theorem in the overall context of the history of ideas and, pursuing Philip Davis’ thoughts, we regard the geometry of triangle as a chapter in the history of mathematical cultures.