Unit VII: Inverse Figures, Circle Systems & Constructibility Criteria — Summary¶
Structural & Heuristic Overview — Inversion, Cirkelsystemer og Konstruktion ved Lineal og Passer (pp. 33–37 & Appendix pp. 89–102 · Problems 192–201 & 403–410)
Architectural Overview (pp. 33–37 & 89–102)¶
Our culminating Unit VII brings together Petersen's highest synthetic machinery—circle inversion, Apollonius/Malfatti circle systems, and algebraic constructibility theory:
Conformal Circle Inversion (
pp. 33–37, Problems 192–201): - Maps \(A \mapsto A_1\) along rays from \(P\) via \(PA \cdot PA_1 = I\), preserving tangency and angles while linearizing circles through \(P\) into straight lines. - Solves Problem 197 (circle through \(P\) tangent to two circles), Problem 199 (circle tangent to three circles concurrent at \(O\), linearized to the incircle/excircles of an inverse triangle), and Castillon’s Problem 200–201.Systems of Circles: Monge, Apollonius & Malfatti (
pp. 89–97, Problems 403–404): - Establishes curvilineal polygon angle sums (pp. 89–92), Monge’s collinearity theorem for external centres of similitude (Theorem 11, p. 93), Gergonne’s pole-and-radical-centre construction for the eight circles of Apollonius’s Problem 403 (p. 95), and Petersen’s celebrated two-page elementary proof of Steiner's construction for Malfatti’s Problem 404 (pp. 95–97).Algebraic Criterion for Straightedge & Compasses (
pp. 97–102, Problems 405–410): - Proves that straightedge-and-compass constructibility holds if and only if sought coordinates lie in an iterated quadratic radical tower over the given data, forcing pencil-intersection curve degrees to be powers \(2^k\). - Demonstrates the impossibility of Pappus’s conchoid (405), general triangle-on-three-circles (407), \((a, c, B-C)\) (408), general angle trisection (409), and Problem 410, while preserving the extraordinary 1879 pre-Lindemann remark (p. 101) that the impossibility of squaring the circle (\(\pi\) not expressible by square roots) had “as yet, nobody been able to prove”!