F.10 Circles and Geometric Mean

Right-Triangle Altitude Mean Proportional, Square Partitions, and Barycentric Center Table

In a semicircle on diameter AB = m + n, the perpendicular altitude h = CH from a point C on the circumference to AB is the geometric mean sqrt(m*n) of the segments AH = m and HB = n, while the radius OC = (m + n)/2 is their arithmetic mean.

Interactive GEOMETOR construction for F.10 (Circles and Geometric 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) G = segment(C, H) A_mean = segment(O, C)

Mathematical Synthesis

By Thales’s theorem and EuclidElements VI.13, any point C on the semicircle with diameter A B forms a right angle \(\angle ACB = 90^\circ\). Dropping the altitude G (\(CH\)) to H on A B creates similar right triangles \(\triangle ACH \sim \triangle CBH\), proving \(CH = \sqrt{AH \cdot HB}\) (geometric mean).

In Topic 10 (SeeligNotebooks), Seelig applies this semicircle geometric mean identity \(P^2 = (r + Q)(r - Q)\) to the three-circle square partition problem and compiles an early summary table of normalized homogeneous barycentric coordinates for \(I, O, H, X(7), X(8), X(6), X(5)\).

Key Algebraic Identities

\[G = |CH| = \sqrt{m n}, \qquad A_{\text{mean}} = |OC| = \frac{m + n}{2}, \qquad H_{\text{mean}} = |CK| = \frac{2 m n}{m + n}\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.10 Circles and Geometric Mean — Folios (Folios 0062–0065, 4 scanned manuscript pages in resources/seelig/10_circles-and-geometric-mean)
Euclid’s Elements	II.14, III.31, VI.8, VI.13 (EuclidElements)
GEOMETOR Elements (G-Index)	G.3 extremities of a segment, G.4 segment
Encyclopedia of Triangle Centers	X(1) — Incenter, X(3) — Circumcenter, X(4) — Orthocenter, X(5) — Nine-Point Center, X(6) — Symmedian (Lemoine) Point, X(7) — Gergonne Point, X(8) — Nagel Point (Kimberling1998)
Seelig Spreadsheets	General Reference, Harmonic Conjugates (Spreadsheets)

Constructive Dependency Graph

Prerequisites: F.8 Chord Intersection, F.9 Circle Equations

Required for: F.17 Ellipse, F.24 Gothic Arch, F.27 Harmonic Mean, F.42 Altitude of a Pythagorean Triple Triangle, F.51 Apollonian Circles and the Problem of Apollonius, F.53 Geometric Half-Angle Trigonometry