Unit VI: Multiplication of Curves & The Method of Similitude — Summary

Structural & Heuristic Overview — Kurvers Multiplikation og Ligedannethedsmethoden (pp. 22–33 · Problems 117–191)

Architectural Overview (pp. 22–33)

Unit VI treats homothety (Multiplication of Curves, pp. 22–27) and the Method of Similitude (pp. 27–33)—which corresponds algebraically to exploiting scale homogeneity by relaxing the single inhomogeneous or boundary condition until the last step:

• Homothetic Multiplication \(\mathcal{H}(P, \pm m/n)\) (pp. 22–27, Problems 117–145): - Scales lines into parallel lines and circles into circles sharing centre of similitude \(P\). - Every pair of unequal circles admits an external (\(+R_1 : R_2\)) and an internal (\(-R_1 : R_2\)) centre of similitude.

• Method of Similitude — Case a: Shape Then Scale (pp. 27–28, Problems 146–153): - When angles, ratios, and only one linear magnitude are given (e.g., Problem 152: triangle from three altitudes \(h_a, h_b, h_c\)), construct any convenient similar prototype first (using \(a : b : c = \frac{1}{h_a} : \frac{1}{h_b} : \frac{1}{h_c}\)) and rescale by fourth-proportional similarity.

• Method of Similitude — Case b: Positional Homothety (pp. 28–33, Problems 154–191): - Suppress one boundary incidence (e.g., in Problem 155, omit \(F \in BC\) when inscribing square \(DEFG\) in \(\triangle ABC\)). All prototype squares resting on \(AB\) with upper-left vertex on \(AC\) are homothetic from \(A\), so their free vertex traces a straight ray \(AF_0\) whose single intersection \(F = AF_0 \cap BC\) solves the problem over \(\mathbb{Q}\) without solving any quadratic equation!