Chapter IX: Inversion — Commentary

Constructive Analysis, Historical Provenance & GEOMETOR Notes (0249–0262, pp. 230–243)

Mathematical Architecture & Conformal Transformation

Chapter IX (§§ 518–546, pp. 230–243) develops circular inversion \((O, k)\) (Steiner, Kelvin, Liouville) as an involutory, angle-preserving transformation of the plane:

  • Images of Lines & Circles (§§ 518–531): Lines/circles through the center of inversion \(O\) invert into straight lines; lines/circles not through \(O\) invert into circles; angles of intersection are preserved in magnitude and reversed in orientation.

  • Peaucellier’s Inversor (§ 524): Exact mechanical straight-line linkage converting circular motion into rectilinear motion via inversion.

  • Crowning Applications (§§ 532–546): Inversion proofs of Ptolemy's theorem and its converse, Steiner’s porism on chains of tangent circles, and the breathtakingly short inversion proof of Feuerbach's theorem (§ 540, Feuerbach 1822) by inverting the nine-point circle, incircle, and an excircle from the midpoint of their common tangent segment!