F.13 Compare Triangles in Circles

Inscribed Triangle Area with Fixed Chord and Circumradius, Nested Radical Chords, and Isosceles Decomposition

For a family of triangles ABC inscribed in a fixed circle of radius R sharing a fixed chord AB = c (with central angle 2*phi where sin(phi) = c/(2R)), the area K decomposes into the signed sum of three central isosceles triangles: K = (1/2) c h_c + (1/2) a h_a - (1/2) b h_b.

Interactive GEOMETOR construction for F.13 (Compare Triangles in Circles). Hover any geometric label in the text or figure to highlight the corresponding element.

O : Point A : Point B : Point C : Point D : Point ===== circle = circle(O, A) AB = segment(A, B) AC = segment(A, C) BC = segment(B, C)

Mathematical Synthesis

Complementing Topic 03 (F.3), Seelig’s 8-folio study in Topic 13 (SeeligNotebooks) tracks how the area \(K = (\triangle ABC)\) and side lengths \(a(\theta; R), b(\theta; R)\) behave as vertex C moves around a fixed circumcircle \(\Gamma_R\) above a fixed chord \(AB = c\).

In Folio 3, he tabulates exact nested-radical chord lengths \(|BC| = \frac{\sqrt{2n^2 \pm 2n\sqrt{n^2-m^2}}}{n}\) for \(c/R = 2m/n \in [1/4, 7/4]\). In Folio 5, he highlights a result marked ‘Surprise & pretty’: decomposing \((\triangle ABC) = \frac{1}{2}ch_c + \frac{1}{2}ah_a - \frac{1}{2}bh_b\) into three isosceles triangles with common apex at the circumcenter O.

Key Algebraic Identities

\[\sin\phi = \frac{c}{2 R}, \qquad h_c = \sqrt{R^2 - \left(\frac{c}{2}\right)^2}, \qquad (ABC) = \frac{1}{2} c h_c + \frac{1}{2} a h_a - \frac{1}{2} b h_b\]

\[a^2 = f^2 + \frac{c^2}{4} + f c \cos\theta, \qquad b^2 = f^2 + \frac{c^2}{4} - f c \cos\theta\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.13 Compare Triangles in Circles — Folios (Folios 0084–0091, 8 scanned manuscript pages in resources/seelig/13_compare-triangles-in-circles)
Euclid’s Elements	III.20, IV.5 (EuclidElements)
GEOMETOR Elements (G-Index)	G.3 extremities of a segment
Encyclopedia of Triangle Centers	X(3) — Circumcenter (Kimberling1998)
Seelig Spreadsheets	Circumcircle Solver, Circumcircle-Circumcenter (Spreadsheets)

Constructive Dependency Graph

Prerequisites: F.3 Biquadratic Forms in Heron’s Formula, F.8 Chord Intersection, F.11 Circumcenter