F.26 Harmonic Conjugate Inversion

Inversion xy = r^2, Steiner’s Power of a Point, the Important Parallelogram MDM_TT, and Diophantine Triples

For a circle Gamma of center M and radius r on diameter AB, internal point C at x = |MC| and its harmonic conjugate D at y = |MD| are inverses satisfying x*y = r^2, while the tangent DT = T forms a parallelogram M D M_T T whose long diagonal z satisfies z^2 = y^2 + 3*r^2 by Stewart’s Theorem.

Interactive GEOMETOR construction for F.26 (Harmonic Conjugate Inversion). 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) P, P’ = tangent_points(D, circle_AB) polar = segment(P, P’)

Mathematical Synthesis

Titled ‘The Relationship Between Harmonic Conjugation, :term:`Inversion`, and The Powers of a Point, as Revealed by An Important Parallelogram’ (FWS 11/19/2013–11/20/2013, SeeligNotebooks), Topic 26 synthesizes inversion (\(x y = r^2\)), Jakob Steiner’s power of a point (\(T^2 = y^2 - r^2 = AD\cdot BD\)), and Stewart's theorem.

Applying Stewart's theorem to the parallelogram \(M D M_T T\) with sides \(r, y\) and short diagonal \(T = |DT|\), Seelig proves that the long diagonal \(z = |M M_T|\) satisfies \(z^2 = y^2 + 3r^2\). Furthermore, by rescaling any primitive Pythagorean triple \((\hat{x}, \hat{t}, \hat{r})\) by \(\hat{x}\), he generates all-integer solutions \((x, t, r, y, T) = (\hat{x}^2, \hat{x}\hat{t}, \hat{x}\hat{r}, \hat{r}^2, \hat{r}\hat{t})\), such as \((9, 12, 15, 25, 20)\) for \(\kappa = 3/5\).

Key Algebraic Identities

\[x y = |MC| \cdot |MD| = r^2, \qquad T^2 = y^2 - r^2 = |AD| \cdot |BD|, \qquad z^2 + T^2 = 2(r^2 + y^2) \implies z^2 = y^2 + 3 r^2\]

\[(x,\, t,\, r,\, y,\, T) = \left(\hat{x}^2,\, \hat{x}\hat{t},\, \hat{x}\hat{r},\, \hat{r}^2,\, \hat{r}\hat{t}\right) \quad \text{where } \hat{x}^2 + \hat{t}^2 = \hat{r}^2\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.26 Harmonic Conjugate Inversion — Folios (Folios 0168–0182, 15 scanned manuscript pages in resources/seelig/26_harmonic-conjugate-inversion)
Euclid’s Elements	III.36, VI.8 (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 (Spreadsheets)

Constructive Dependency Graph

Prerequisites: F.8 Chord Intersection, F.25 Harmonic Conjugates