F.27 Harmonic Mean

Compass Construction of the Harmonic Mean, Prime-Number Harmonic Ranges, and the Bilinear Lens

If C and D are harmonic conjugates with respect to segment AB (so |AC|/|CB| = |AD|/|BD|), then the length |CD| is the harmonic mean of |BD| and |AD|: |CD| = 2 |BD| |AD| / (|BD| + |AD|) = r (gamma + 1/gamma).
ABC semicircle G A_mean H_mean A (0.000, 0.000) H (5.000, 0.000) B (7.000, 0.000) O (3.500, 0.000) C (5.000, 3.162) K (3.776, 0.581) AA HH BB OO CC KK + − ⟲ Lbl Gde

Interactive GEOMETOR construction for F.27 (Harmonic Mean). Hover any geometric label in the text or figure to highlight the corresponding element.

A : Point B : Point H : Point ===== O = midpoint(A, B) semicircle = circle(O, B) C = perpendicular_chord(H, AB, semicircle) K = foot(H, segment(O, C)) H_mean = segment(C, K)

Mathematical Synthesis

Dated October 25 and November 13–14, 2013 (SeeligNotebooks), Topic 27 develops both synthetic compass-and-straightedge constructions and algebraic number theory for the harmonic mean. Citing Heath’s notes on EuclidElements VI.3 (p. 198), Seelig shows that for harmonic points \((A, B; C, D)\) with \(|BD| = r(\gamma - 1)\) and \(|AD| = r(\gamma + 1)\), the segment \(|CD| = r(\gamma + 1/\gamma)\) is the harmonic mean of \(|BD|\) and \(|AD|\), and gives a 7-step compass construction laying \(n = |BD|\) onto \(m = |AD|\) to form circle \(\Gamma_0\) of diameter \(m - n\).

In Folios 15–16 (Problems 6–10), he studies harmonic ranges whose adjacent segments \(X = AC, Y = CB\) are consecutive prime numbers \(P_{n+1}, P_n\), finding \(Z = \frac{P_n(P_n+P_{n+1})}{P_{n+1}-P_n}\) (\(10, 12, 30, 63/2\)), and in Folios 17–19 (‘Bilinear Transform As Viewed Through the Lens of :term:`Harmonic Conjugates <harmonic conjugate>`’) explains why \(\kappa = x/r \in (0, 1)\) is algebraically more fundamental than Apollonius’s ratio \(k \in (1, \infty)\).

Key Algebraic Identities

\[|CD| = \operatorname{HM}(|BD|,\, |AD|) = \frac{2 |BD| \cdot |AD|}{|BD| + |AD|} = r\left(\gamma + \frac{1}{\gamma}\right), \qquad x^2 = \frac{(r / t)^2}{\frac{1}{r^2} + \frac{1}{t^2}} = \left(\frac{r}{\gamma}\right)^2\]
\[Y^2 + (X + Z) Y - X Z = 0 \implies Y = \frac{-(X + Z) + \sqrt{(X + Z)^2 + 4 X Z}}{2}\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios

F.27 Harmonic Mean — Folios (Folios 0183–0201, 19 scanned manuscript pages in resources/seelig/27_harmonic-mean)

Euclid’s Elements

VI.3, VI.8, VI.13 (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

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Prerequisites: F.2 Bilinear Transformations, F.10 Circles and Geometric Mean, F.25 Harmonic Conjugates

Required for: F.44 Homothetic Ratios and Crossed Diagonals, F.51 Apollonian Circles and the Problem of Apollonius