F.12 Coaxial Circles

Equipotential Pencils Around Point Charges, Limiting Points, and Orthogonal Families

A hyperbolic pencil of coaxial circles with limiting points L_1 = (-r, 0) and L_2 = (+r, 0) coincides with the Apollonian equipotential circles d_1 / d_2 = k around two point charges Q_1, Q_2, sharing the y-axis as their common radical axis.

Interactive GEOMETOR construction for F.12 (Coaxial Circles). Hover any geometric label in the text or figure to highlight the corresponding element.

L_1 : Point L_2 : Point R_1 : Point R_2 : Point ===== radical_axis = line(R_1, R_2) line_of_centers = line(L_1, L_2) C_1 = coaxial_circle(L_1, L_2, -3.5) C_2 = coaxial_circle(L_1, L_2, 2.5)

Mathematical Synthesis

Inspired by electrostatic equipotential contours around fixed point charges \(Q_1(-r, 0)\) and \(Q_2(+r, 0)\) in Kraus’s Electromagnetics and C. Stanley Ogilvy’s Excursions in Geometry (Ogilvy1969), Seelig’s 12-folio study (Friday April 24, 2015, SeeligNotebooks) unifies coaxial circles, harmonic conjugate division, radical axis theory, and Apollonian circles.

For potential ratio \(k = Q_1/Q_2 = d_1/d_2 = 2\), Seelig determines six landmark points \(P_1,\dots,P_6\) on the equipotential circle: the internal and external harmonic division points \(P_1 = (\frac{1}{3}r, 0)\) and \(P_2 = (3r, 0)\), center \(M' = (\frac{5}{3}r, 0)\), radius \(R = \frac{4}{3}r\), vertical chord endpoints \(P_3, P_4 = (r, \pm \frac{2}{\sqrt{3}}r)\), and right-triangle vertices \(P_5, P_6\) on the circle \(x^2 + y^2 = 2r^2\).

Key Algebraic Identities

\[P_1 = \left(\frac{k - 1}{k + 1}r,\, 0\right), \qquad P_2 = \left(\frac{k + 1}{k - 1}r,\, 0\right), \qquad M' = \frac{k^2 + 1}{k^2 - 1}r, \qquad R = \frac{2 k}{k^2 - 1}r\]

\[x^2 + y^2 - 2 c_x x + r^2 = 0, \qquad E_{\text{TOT}} = k Q_1 \frac{x - x_1}{|x - x_1|^3} + k Q_2 \frac{x - x_2}{|x - x_2|^3} = 0\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.12 Coaxial Circles — Folios (Folios 0072–0083, 12 scanned manuscript pages in resources/seelig/12_coaxial-circles)
Euclid’s Elements	III.35, III.36 (EuclidElements)
GEOMETOR Elements (G-Index)	G.3 extremities of a segment
Encyclopedia of Triangle Centers	General projective / metric locus (Kimberling1998)
Seelig Spreadsheets	Coaxial Circles, Coaxal Circle Table, Apollonian Circles, Radical Axes (Spreadsheets)

Constructive Dependency Graph

Prerequisites: F.2 Bilinear Transformations, F.8 Chord Intersection

Required for: F.25 Harmonic Conjugates, F.43 Radical Axes and Radical Center, F.51 Apollonian Circles and the Problem of Apollonius