The Art of Inversion: Turning Circles into Lines and Difficulty Figures into Familiar Ones

Inversion is the great magic trick of geometry — and this book teaches you to work it. A single rule turns the inside of a circle into its outside, bends straight lines into circles, and collapses crowded, impossible-looking figures into calm, familiar ones. The theorem you could not see was there all along, wearing a disguise. The Art of Inversion shows you how to remove the disguise. This is not a formula sheet with problems attached. It is a discovery-driven course in the geometry of circles, built the way mathematics is actually found: every chapter opens with a problem you cannot yet solve, and the theory is built only once you have felt the need for it. You predict before you calculate, and you earn each theorem instead of being handed it. Written for ambitious students of competition and olympiad mathematics — and for the teachers who coach them — the book carries the reader from the very definition of inversion to the frontier of non-Euclidean geometry, one well-chosen mirror at a time. Inside you will find:

• The complete inversive toolkit: the generalized-circle dictionary, orthogonality and coaxal pencils, cross-ratio, and the algebraic complex-number mirror.
• The classical engines of olympiad geometry — Ptolemy and Casey, Apollonius' tangency problems, Descartes' circle theorem, the arbelos and Steiner chains.
• Inversion inside the triangle: the incircle touchpoint, symmedians, mixtilinear circles, and a guided project on Feuerbach's theorem.
• The great classics rebuilt one inversion at a time: Miquel, Simson, Steiner, the Butterfly, and the Japanese temple-tablet theorem.
• New worlds — stereographic projection on the sphere and the Poincaré disk, where inversion becomes the reflection of a whole new geometry.
• A strategist's workbook, a clinic of flawed proofs to diagnose and repair, and a gallery of long, ambitious problems.

Roughly one hundred clean, purpose-built diagrams make every construction visible, and hundreds of Review and Challenge problems — each with a one-line nudge rather than a spoiler — turn reading into doing. Discover inversion not as a trick, but as a way of seeing. The Art of Inversion is a volume in The Art of Creative Math, a series that develops one powerful mathematical idea at a time through intuition, elegant reasoning, and problems worth exploring.