Unit IV: Chapter II, Parts A & B — Parallel Translation & Replacing — Summary

Structural & Heuristic Overview — Kapitel II — Figurens Omdannelse: Parallelforskydning og Omlægning (pp. 46–60 · Problems 246–335)

Architectural Overview (pp. 46–60)

Chapter II inaugurates Transformation of the Figure (Figurens Omdannelse): when given lengths and angles lie dispersed across a figure so no constructible sub-triangle exists immediately, we apply an invertible geometric transformation to gather the scattered data into one auxiliary triangle, construct that auxiliary figure, and invert the transformation.

Four Paradigms of Parallel Translation & Three Modes of Replacing

• A. Parallel Translation (pp. 46–55, Problems 246–310): 1. Triangle Medians (p. 47, Problem 246): Shifting sides/medians forms an auxiliary triangle with sides \((2m_a, 2m_b, 2m_c)\) (or \((\frac{2}{3}m_a, \frac{2}{3}m_b, \frac{2}{3}m_c)\)), turning median problems into standard triangle problems. 2. Quadrilateral & Trapezoid Gathering (pp. 48–49, Problem 259): Translating two adjacent sides of \(ABCD\) builds the diagonal parallelogram \(BD_1\) (area \(2[ABCD]\)); in a trapezoid (\(CD \parallel AB\)), translating leg \(BC\) parallel to \(CD\) into \(ED\) concentrates all four given sides \((a, b, c, d)\) into the SSS triangle \(\triangle AED\) of sides \((a - b, c, d)\). 3. Vector Bridge Between Two Curves (pp. 51–52, Problems 271–274): Placing segment \(XY \parallel \vec{m}\) of length \(|\vec{m}|\) with \(X \in K_1, Y \in K_2\) reduces to intersecting \((K_1 + \vec{m}) \cap K_2\). 4. Shrinking a Circle to a Point (pp. 52–53, Problems 275–278): Subtracting the smallest radius \(r\) reduces circle-circle and line-circle tangency problems to tangents through a point!

• B. Replacing (pp. 56–60, Problems 311–335): 1. Gathering opposite chord data by chord reversal in a circle (Problem 311); 2. Introducing angle differences \(A - B\) by flipping a triangle across its perpendicular bisector (Problem 313); 3. Making unknown proportional lines cover each other by scaling and flipping \(\triangle ABC\) onto \(\triangle ADC_1\), solving the cyclic quadrilateral from four sides (Problem 316).