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Unit II: Chapter I, Part A — Loci of Points & The Method of Two Loci — Notes¶

Historical, Mathematical & Cross-Repository Commentary (pp. 7–22, 37–40)

Historical, Mathematical & GEOMETOR Commentary¶

1. Projective Harmony Behind Petersen’s Loci `f` and `g`¶

Petersen’s juxtaposition of Locus f (Circle of Apollonius, pp. 11–12) and Locus g (harmonical rays, p. 12) quietly introduces projective harmonic division into elementary compass-and-straightedge geometry:

For any \(m \neq n\), the internal and external ratio points \(C\) and \(C_1\) on \(AB\) satisfy the signed harmonic cross-ratio \((A, B; C, C_1) = -1\).
Because internal and external bisectors \(PC\) and \(PC_1\) of \(\angle APB\) are always orthogonal (\(\text{spread}(PC, PC_1) = 1\)), Euclid VI.3 converts the metric ratio \(PA : PB = m : n\) into Thales’s right-angle locus (e) on fixed diameter \(CC_1\)!
In Problem 22 (p. 14), intersecting two Apollonius circles of \(\triangle ABC\) for ratios inversely proportional to the sides (\(PA \cdot a = PB \cdot b = PC \cdot c\)) constructs the first and second isodynamic points X(15) and X(16) in Kimberling’s Encyclopedia (geometor/etc).

2. Radical Axis & Coaxal Pencils (Locus `h`, Problem 30)¶

Notice how effortlessly Petersen derives Gaspard Monge's and Louis Gaultier’s radical axis (Locus h, App. 1, p. 13) without coordinates: equating tangent squares \(t_1^2 = PA^2 - r_1^2\) and \(t_2^2 = PB^2 - r_2^2\) immediately reduces equal power to \(PB^2 - PA^2 = r_2^2 - r_1^2 = \text{const.}\)—which is already proven in Locus h via the Pythagorean projection theorem to be a straight line perpendicular to \(AB\)! Transitivity of equality (\(t_1 = t_2\) and \(t_2 = t_3 \implies t_1 = t_3\)) then yields the concurrency of the three radical axes at the radical centre (Problem 30, p. 14) in two lines.

3. Exact `geometor.model` Realization: `apollonius_two_loci`¶

In geometor.petersen.constructions.apollonius_two_loci(), choosing \(A = (0, 0)\), \(B = (15, 0)\), \(m : n = 2 : 1\), and altitude \(h = 8\) yields:

Harmonic conjugates \(C = (10, 0)\) and \(C_1 = (30, 0)\) with cross_ratio(A, B, C, C1) == -1;
Apollonius circle centered at \(M = (20, 0)\) of radius \(R = 10\);
Two integer intersection solutions \(P_1 = (14, 8)\) and \(P_2 = (26, 8)\) where \(Q(P_1, A) = 260\), \(Q(P_1, B) = 65\) (\(260 = 4 \cdot 65\)), and line_spread([P1, C], [P1, C1]) == 1.

Cross-Repository Concordance Links¶

(See the full Petersen Master Concordance for complete tables, and demos for executable SymPy demonstrations.)

Chapter I.A, Loci a–e (pp. 7–11) (Problems 1, 3, 16, 18, 19) (Elementary Point Loci & the Snellius–Pothenot Three-Point Problem (Chapter I.A, Loci a–e (pp. 7–11))): Systematizes the circle (a), parallel pair (b), perpendicular bisector (c), angle bisectors (d), and subtended-angle circular arcs (e), culminating in oriented inscribed angles and Pothenot’s geodesic resection (Problem 19). - : Provides the exact Euclidean propositions underlying Petersen’s Loci c, e, and tangent Problem 16. - : Computable compass-and-straightedge primitives invoked inside Petersen’s locus intersections.
Chapter I.A, Loci f & g (pp. 11–12) (Problems 20, 21, 22, 45, 47, 48) (Circle of Apollonius (PA : PB = m : n) & Harmonical Rays (Chapter I.A, Loci f & g (pp. 11–12))): Proves via internal and external angle bisectors PC, PC1 (Euclid VI.3) that PA : PB = m : n traces the circle on harmonic diameter CC1 where (A, B; C, C1) = -1, dualized for two lines in Locus g (harmonical rays). - : Develops Petersen’s Problem 22 (finding a point whose distances from the three vertices of a triangle have given ratios -> X(15), X(16)) into a full section. - : Expands Petersen’s harmonic diameter CC1 and Locus g (harmonical rays) into projective harmonic ranges and pencils. - geometor/pappus — Complete Quadrangle & Harmonic Conjugates: Projective invariance of the harmonic cross-ratio (A, B; C, C1) = -1. - : Pairwise intersections of the three Apollonius circles of a triangle (Petersen Problem 22).
Chapter I.A, Loci h, i, k (pp. 12–14) (Problems 24, 30, 31, 32, 37, 38) (Difference & Sum of Squares, Radical Axis, Radical Centre & Centroid (Chapter I.A, Loci h, i, k (pp. 12–14))): Derives the perpendicular line locus PB^2 - PA^2 = a^2 (h), radical axis and radical centre of three circles (Problem 30), median/Apollonius theorem locus PA^2 + PB^2 = a^2 (i), and area-trisecting centroid proof (Problem 32). - : Expands Petersen’s Locus h App. 1–3 and Problem 30 into the complete synthetic theory of coaxal pencils. - : Triangle-center realizations of Petersen’s Problem 32 and Problem 30.