F.16 d’Alembert’s Theorem

Monge–d’Alembert Collinearity of the Six Centers of Similitude of Three Circles

Given three circles of distinct radii in the plane, their six pairwise internal and external centers of similitude lie by threes on four projective lines L_{+++}, L_{–+}, L_{-+-}, and L_{+–} (the Monge–d’Alembert similitude axes).

Interactive GEOMETOR construction for F.16 (d’Alembert’s Theorem). Hover any geometric label in the text or figure to highlight the corresponding element.

O_1 : Point R_1 : Point O_2 : Point R_2 : Point O_3 : Point ===== C_1 = circle(O_1, R_1) C_2 = circle(O_2, R_2) C_3 = circle(O_3, R_3) S_12 = external_similitude(C_1, C_2) S_23 = external_similitude(C_2, C_3) dAlembert_axis = line(S_12, S_23)

Mathematical Synthesis

Building on Topic 05 (F.5), Seelig’s multi-color constructions in Topic 16 (SeeligNotebooks) illustrate the theorem of d'Alembert and Gaspard Monge: for three circles \(\Gamma_A, \Gamma_B, \Gamma_C\), the three external centers of similitude \(S_{\text{EX }AB}, S_{\text{EX }AC}, S_{\text{EX }BC}\) are collinear on axis \(L_{+++}\), while each external center is collinear with the two complementary internal centers on \(L_{--+}, L_{-+-}, L_{+--}\).

In Folios 3–4, Seelig connects these four similitude axes to Gergonne’s solution of the Problem of :person:`Apollonius <apollonius>` (finding the 8 circles simultaneously tangent to \(\Gamma_A, \Gamma_B, \Gamma_C\)) and executes a compass construction of five mutually tangent circles (inner and outer Soddy circles).

Key Algebraic Identities

\[L_{+++}:\; \{S_{\text{EX}\,AB},\, S_{\text{EX}\,BC},\, S_{\text{EX}\,CA}\}, \qquad L_{+--}:\; \{S_{\text{EX}\,AB},\, S_{\text{IN}\,BC},\, S_{\text{IN}\,CA}\}\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.16 d’Alembert’s Theorem — Folios (Folios 0114–0117, 4 scanned manuscript pages in resources/seelig/16_dalembert)
Euclid’s Elements	VI.2, VI.4 (EuclidElements)
GEOMETOR Elements (G-Index)	G.3 extremities of a segment
Encyclopedia of Triangle Centers	General projective / metric locus (Kimberling1998)
Seelig Spreadsheets	Centers of Similitude (Spreadsheets)

Constructive Dependency Graph

Prerequisites: F.5 Centers of Similitude