F.9 Circle Equations

Three-Circle Distance Systems in a Square and a Sangaku Tangent Circle Problem

Pairwise subtraction of Cartesian circle equations (x - u_i)^2 + (y - v_i)^2 = r_i^2 eliminates the quadratic terms x^2 + y^2 to yield linear radical-axis relations, determining both the side Z of a square from three corner distances and the radius of a Sangaku circle tangent to an inscribed square and equilateral triangle.

Interactive GEOMETOR construction for F.9 (Circle Equations). Hover any geometric label in the text or figure to highlight the corresponding element.

O : Point A : Point B : Point C : Point D : Point ===== circle = circle(O, A) AB = segment(A, B) CD = segment(C, D) P = intersect(AB, CD)

Mathematical Synthesis

In Topic 09 (SeeligNotebooks), Seelig examines two multi-circle systems in Cartesian coordinates. In Folios 1–4 (FWS 9/24/2014), an interior point \(P(P, Q)\) inside a square \(ABCD\) of side \(Z\) has distances \(PA = 3, PB = 5, PD = 7\). Subtracting the three circle equations yields \(-2ZP + Z^2 = 40\), \(-2ZQ + Z^2 = 16\), and \(2Z(Q - P) = 24\).

In Folios 5–8 (Posted Sept. 28, 2014), Seelig models a Sangaku problem posed by Tony Garcia: inside a circle \(\Gamma_O(R)\) containing a rotated square \(ACBD\) of side \(\sqrt{2}R\), an equilateral triangle \(\triangle ACE\), and circular arcs \(\widehat{AB}\) and \(\widehat{CD}\), he derives the center \(P_y = \frac{r_F^2 - r_I^2}{2(F_y - I_y)}\) and radius \(r\) of the inscribed circle tangent to both the triangle sides and the circular arcs.

Key Algebraic Identities

\[x^2 + y^2 = 9, \quad x^2 + (y - Z)^2 = 25, \quad (x - Z)^2 + y^2 = 49 \implies 2 Z (Q - P) = 24\]

\[l_{AE}:\; y = \sqrt{3}x - \frac{\sqrt{2}}{2}R, \qquad P_y = \frac{r_F^2 - r_I^2}{2(F_y - I_y)}\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.9 Circle Equations — Folios (Folios 0054–0061, 8 scanned manuscript pages in resources/seelig/09_circle-equations)
Euclid’s Elements	I.47, III.35 (EuclidElements)
GEOMETOR Elements (G-Index)	G.3 extremities of a segment
Encyclopedia of Triangle Centers	General projective / metric locus (Kimberling1998)
Seelig Spreadsheets	General Reference, Radical Axes (Spreadsheets)

Constructive Dependency Graph

Prerequisites: F.8 Chord Intersection

Required for: F.10 Circles and Geometric Mean, F.24 Gothic Arch, F.43 Radical Axes and Radical Center