F.5 Centers of Similitude

Internal and External Homothetic Centers Dividing the Line of Centers Harmonically

Given two circles (O_1, r_1) and (O_2, r_2) separated by center distance d = |O_1 O_2| with r_2 > r_1, their common internal and external tangents concur at the internal and external centers of similitude S_IN and S_EX, which divide O_1 O_2 harmonically in the ratio r_1 : r_2.
C_1 C_2 axis tangent_ext r_1 r_2 O_1 (0.000, 0.000) R_1 (0.000, 1.500) O_2 (5.000, 0.000) R_2 (5.000, 3.000) S_+ (1.667, 0.000) S_- (-5.000, 0.000) O1O1 R1R1 O2O2 R2R2 S+S+ S-S- + − ⟲ Lbl Gde

Interactive GEOMETOR construction for F.5 (Centers of Similitude). 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 ===== C_1 = circle(O_1, R_1) C_2 = circle(O_2, R_2) S_IN = internal_similitude(O_1, R_1, O_2, R_2) S_EX = external_similitude(O_1, R_1, O_2, R_2)

Mathematical Synthesis

Citing Volume I, p. 207 of his geometry notebooks (SeeligNotebooks), Seelig constructs the centers of similitude of two circles C1 \((O_1, r_1)\) and C2 \((O_2, r_2)\) separated by \(d = |O_1 O_2|\). The internal center SIN (\(S_{IN}\)) and external center SEX (\(S_{EX}\)) divide the segment \(O_1 O_2\) internally and externally in the exact ratio \(r_1 : r_2\), forming a four-point harmonic conjugate range \((O_1, O_2; S_{IN}, S_{EX})\).

Key Algebraic Identities

\[|O_1 S_{IN}| = d \frac{r_1}{r_2 + r_1}, \qquad |S_{IN} O_2| = d \frac{r_2}{r_2 + r_1}\]
\[|S_{EX} O_1| = d \frac{r_1}{r_2 - r_1}, \qquad |S_{EX} O_2| = d \frac{r_2}{r_2 - r_1}\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios

F.5 Centers of Similitude — Folios (Folios 0021–0021, 1 scanned manuscript page in resources/seelig/05_centers-of-similitude)

Euclid’s Elements

VI.2, VI.4 (EuclidElements)

GEOMETOR Elements (G-Index)

G.3 extremities of a segment

Encyclopedia of Triangle Centers

X(55) — Internal Center of Similitude of Incircle and Circumcircle, X(56) — External Center of Similitude of Incircle and Circumcircle (Kimberling1998)

Seelig Spreadsheets

Centers of Similitude (Spreadsheets)

Constructive Dependency Graph

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Prerequisites: F.2 Bilinear Transformations

Required for: F.16 d’Alembert’s Theorem, F.20 Extangent, F.44 Homothetic Ratios and Crossed Diagonals, F.48 Verisimilitude, Corner Subtriangles, and Excircle Homothety