F.25 Harmonic Conjugates

Orthogonal Circles, Ogilvy’s Theorem, Dual Relations, and the Pi_+/Pi_- Symmetry Master Diagram

Collinear points C and D divide segment AB (of midpoint M and half-length r) internally and externally in the same ratio k = |AC|/|BC| = |AD|/|BD| if and only if |MC|*|MD| = r^2, which occurs if and only if the circle on diameter AB and the Apollonian circle on diameter CD intersect orthogonally.

Interactive GEOMETOR construction for F.25 (Harmonic Conjugates). Hover any geometric label in the text or figure to highlight the corresponding element.

A : Point B : Point C : Point ===== M = midpoint(A, B) circle_AB = circle(M, B) D = harmonic_conjugate(A, B, C) O_A = midpoint(C, D) Apollonius = circle(O_A, C) P, P’ = intersect(circle_AB, Apollonius) polar = segment(P, P’)

Mathematical Synthesis

Spanning 28 folios (FWS 6/20/2014, SeeligNotebooks) and citing C. Stanley Ogilvy (Ogilvy1969, pp. 19–27) and Heath’s commentary on EuclidElements VI.3 (p. 198), Topic 25 is Seelig’s definitive treatise on harmonic conjugate geometry and orthogonal circles.

Setting \(\kappa = |MC|/r = \frac{k-1}{k+1}\) on circle circleAB (center M, radius \(r\)) and \(\kappa' = |M'B|/R = 1/k\) on the orthogonal Apollonius circle (center OA, radius \(R\)), Seelig proves his Theorem 1 (Duality Theorem): \((\kappa, k)\) and \((\kappa', k')\) are reciprocal duals (\(\kappa' = 1/k, k' = 1/\kappa\)). Introducing the hyperbolic-like invariants \(\Pi_+ \triangleq \frac{1}{\kappa} + \kappa\) and \(\Pi_- \triangleq \frac{1}{\kappa} - \kappa\) (satisfying \(\Pi_+^2 - \Pi_-^2 = 4\)), he expresses every chord, altitude \(p = PQ\), projection \(q = MQ\), center distance \(d = MM'\), and tangent \(T, T'\) in closed symmetric form, verified with the exact rational example \(\kappa = 3/7, r = 42, R = 40, d = 58\).

Key Algebraic Identities

\[\frac{|AC|}{|CB|} = \frac{|AD|}{|BD|} = k = \frac{1 + \kappa}{1 - \kappa}, \qquad |MC| \cdot |MD| = x y = r^2, \qquad \frac{2}{|AB|} = \frac{1}{|AC|} + \frac{1}{|AD|}\]

\[\Pi_+ = \frac{1}{\kappa} + \kappa, \quad \Pi_- = \frac{1}{\kappa} - \kappa, \quad \Pi_+^2 - \Pi_-^2 = 4, \quad R = \frac{r}{2}\Pi_-, \quad d = \frac{r}{2}\Pi_+, \quad p = r\frac{\Pi_-}{\Pi_+}\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.25 Harmonic Conjugates — Folios (Folios 0140–0167, 28 scanned manuscript pages in resources/seelig/25_harmonic-conjugates)
Euclid’s Elements	III.36, VI.3 (EuclidElements)
GEOMETOR Elements (G-Index)	G.3 extremities of a segment, G.5 surface
Encyclopedia of Triangle Centers	General projective / metric locus (Kimberling1998)
Seelig Spreadsheets	Harmonic Conjugates, HC Tables, Apollonian Circles, Coaxial Circles (Spreadsheets)

Constructive Dependency Graph

Prerequisites: F.2 Bilinear Transformations, F.8 Chord Intersection, F.12 Coaxial Circles

Required for: F.26 Harmonic Conjugate Inversion, F.27 Harmonic Mean, F.31 Logic Tests, F.39 Pole and Polar with Respect to a Triangle, F.44 Homothetic Ratios and Crossed Diagonals, F.51 Apollonian Circles and the Problem of Apollonius