F.43 Radical Axes and Radical Center

Two-Circle Radical Division, Vertex-Altitude Circle Radical Center, and Circular Lens Regimes

The radical axis of two circles of radii r_1, r_2 with center separation d cuts the line of centers at x = d/2 + (r_1^2 - r_2^2)/(2 d) from O_1, and the three radical axes of the vertex-altitude circles Gamma_A(h_A), Gamma_B(h_B), Gamma_C(h_C) concur at the radical center W.

Interactive GEOMETOR construction for F.43 (Radical Axes and Radical Center). Hover any geometric label in the text or figure to highlight the corresponding element.

O_1 : Point A_1 : Point O_2 : Point A_2 : Point ===== c_1 = circle(O_1, A_1) c_2 = circle(O_2, A_2) l_centers = line(O_1, O_2) P_1 = intersection(c_1, c_2) rad_axis = perpendicular(P_1, l_centers)

Mathematical Synthesis

Spanning 11 folios (SeeligNotebooks), Seelig’s treatise on the radical axis begins (Folio 1) with the fundamental two-circle division formula \(x = \frac{d}{2} + \frac{r_1^2 - r_2^2}{2d}\), tested on \(r_1=1, r_2=2, d=5\) to give \(|O_1 P| = 2.2, |O_2 P| = 2.8\), and equal tangent length \(t = \frac{4\sqrt{6}}{5} \approx 1.9596\). Folios 2–8 (FWS Dec. 8/2014) execute an ambitious 7-page analytic calculation: placing circles \(\Gamma_A(h_A), \Gamma_B(h_B), \Gamma_C(h_C)\) at the vertices of a triangle with radii equal to the three altitudes, Seelig derives each pairwise radical axis in double-intercept form \(\frac{x}{x_0} + \frac{y}{y_0} = 1\) and solves for their common radical center \(W = (2.586, 0.603)\) on the \((2,3,4)\) triangle.

Folios 9–11 classify the five relative position regimes of two circles as \(d\) varies from \(d > r_1 + r_2\) (disjoint) through external/internal tangency to \(d < |r_1 - r_2|\), recording the circular lens intersection chord and area.

Key Algebraic Identities

\[x = |O_1 P| = \frac{d}{2} + \frac{r_1^2 - r_2^2}{2 d}, \qquad x' = |O_2 P| = \frac{d}{2} - \frac{r_1^2 - r_2^2}{2 d}, \qquad t^2 = x^2 - r_1^2 = {x'}^2 - r_2^2\]

\[\Gamma_A(A, h_A),\; \Gamma_B(B, h_B),\; \Gamma_C(C, h_C) \implies \frac{x}{x_0} + \frac{y}{y_0} = 1\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.43 Radical Axes and Radical Center — Folios (Folios 0271–0281, 11 scanned manuscript pages in resources/seelig/43_radical-axes)
Euclid’s Elements	II.5, III.36 (EuclidElements)
GEOMETOR Elements (G-Index)	G.7 plane surface, G.11 rectilineal figures
Encyclopedia of Triangle Centers	General projective / metric locus (Kimberling1998)
Seelig Spreadsheets	Radical Axes, Coaxial Circles (Spreadsheets)

Constructive Dependency Graph

Prerequisites: F.9 Circle Equations, F.12 Coaxial Circles