Unit V: Axial Reflection & Chapter III — The Theory of Revolution — Summary

Structural & Heuristic Overview — Drejning om en Axe og Kapitel III — Drejningstheorien (O, v, f) (pp. 60–68 · Problems 336–362)

Architectural Overview (pp. 60–68)

Unit V pairs the culmination of Chapter II (Revolution Around an Axis, i.e., axial reflection, pp. 60–64) with Chapter III (The Theory of Revolution, i.e., planar spiral similarity around a pivot point, pp. 65–68):

• Axial Reflection (pp. 60–64, Problems 336–348): - Straightens broken equal-angle trajectories across mirror lines: Hero’s shortest path (Problem 338), n-cushion polygonal billiards (Problem 339), and minimum-perimeter inscribed polygons (Problem 340, contrasting determinate odd-\(n\) polygons such as the orthic triangle with porismatic/impossible even-\(n\) polygons).

• Planar Spiral Revolution \(\mathcal{R}(O, v, f)\) (pp. 65–68, §§ 1–5 & Problems 349–362): - Rotates every ray \(Oa\) through angle of revolution \(v\) and scales its length by ratio of revolution \(f = OA / Oa\) around pivot point \(O\). - Fundamental Placement Principle (§ 4, p. 66): To place \(\triangle OQR\) of given shape with \(O\) fixed, \(Q \in k_1\), and \(R \in k_2\), revolve \(k_1\) by \((O, \angle QOR, OR/OQ)\) into \(K_1\) and intersect \(K_1 \cap k_2\)! - Equilateral Specializations (v = 60°, f = 1): Solves placing an equilateral triangle on three parallel lines (Problem 349) and on three concentric circles (Problem 350) in two compass-and-straightedge steps. - Miquel Pivot Theorem (§ 5, p. 68): Proves that any two directly similar non-translated figures possess a unique finite pivot point \(O\), constructed as the second intersection of circles \((PAa)\) and \((PBb)\) where \(P = AB \cap ab\).