F.51 Apollonian Circles and the Problem of Apollonius

Locus |PA|/|PB| = k, Integer Mean Triples, Power of a Point, and Dörrie’s 8-Circle Tangency Table

The locus of points P with distance ratio |PA| / |PB| = k > 1 (where |AB| = d) is the Apollonian circle of radius R = d k / (k^2 - 1) centered at distance M’ = d / (k^2 - 1) beyond B, cutting line AB at the internal and external harmonic conjugates C and D such that |A T’|^2 = |AC| |AD| = |AB| |A M’|.

Interactive GEOMETOR construction for F.51 (Apollonian Circles and the Problem of Apollonius). Hover any geometric label in the text or figure to highlight the corresponding element.

A : Point B : Point C : Point ===== D = harmonic_conjugate(A, B, C) M_p = midpoint(C, D) c_apol = circle(M_p, C)

Mathematical Synthesis

Dated February 7, 2014 (SeeligNotebooks), Seelig’s nine-folio treatise on the Apollonian circles combines the locus definition \(|PA|/|PB| = k\) (Folios 1–8) with the classic Problem of :person:`Apollonius` (Folio 9, citing Dorrie’s 100 Great Problems of Elementary Mathematics, Problem #32, pp. 154–160, Dorrie1965). Given \(|AB| = d\), the internal and external bisectors of \(\angle APB\) meet \(AB\) at the harmonic conjugate points C and D (Euclid Elements VI.3), giving diameter endpoints separated by \(2R = \frac{2dk}{k^2-1}\) and center Mp at \(M' = \frac{d}{k^2-1}\) from B.

Across Folios 4–7, Seelig proves the power-of-a-point tangent identity \(|AT'|^2 = |AC|\cdot|AD| = |AB|\cdot|AM'|\) and asks when two numbers \(m, n\) yield simultaneous integer arithmetic, geometric, and harmonic means—finding the parametric solution \(m = 36, n = 324 \implies A = 180, G = 108, H = 64.8\) (or scaled by 5 for all integers). Folio 9 classifies all \(2^3 = 8\) Apollonian tangent circles to three given circles by their internal (I) or external (X) tangency signatures.

Key Algebraic Identities

\[\frac{|PA|}{|PB|} = k \implies M' = \frac{d}{k^2 - 1}, \qquad R = \frac{d k}{k^2 - 1}, \qquad R^2 - {M'}^2 = \frac{d^2}{k^2 - 1}\]

\[|A T'|^2 = |AC|\cdot|AD| = |AB|\cdot|A M'| = d(d + M') = \frac{d^2 k^2}{k^2 - 1}\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.51 Apollonian Circles and the Problem of Apollonius — Folios (Folios 0322–0330, 9 scanned manuscript pages in resources/seelig/51_apollonian-circles)
Euclid’s Elements	III.36, VI.3 (EuclidElements)
GEOMETOR Elements (G-Index)	G.5 surface, G.7 plane surface
Encyclopedia of Triangle Centers	General projective / metric locus (Kimberling1998)
Seelig Spreadsheets	Apollonian Circles, Harmonic Conjugates, Coaxial Circles (Spreadsheets)

Constructive Dependency Graph

Prerequisites: F.2 Bilinear Transformations, F.10 Circles and Geometric Mean, F.12 Coaxial Circles, F.25 Harmonic Conjugates, F.27 Harmonic Mean