F.45 Squares, Subdivided Lattices, and the Japanese Theorem

Inscribed Squares, Cevians Inside a Square, Equilateral Subdivisions, and Carnot’s Theorem

A square inscribed in a circle of radius R has side s = sqrt(2) R; inside a square of side s, intersecting cevians divide the interior in exact rational proportions, and by Carnot’s Theorem the sum of inradii in any triangulation of a cyclic polygon is invariant.

Interactive GEOMETOR construction for F.45 (Squares, Subdivided Lattices, and the Japanese Theorem). Hover any geometric label in the text or figure to highlight the corresponding element.

O : Point A : Point ===== c_out = circle(O, A) l_oa = line(O, A) C = intersection(l_oa, c_out) perp = perpendicular(O, l_oa) B = intersection(perp, c_out) D = intersection(perp, c_out)

Mathematical Synthesis

This six-folio compilation (SeeligNotebooks) brings together four geometric studies centered on regular polygons and triangulations: Folio 1 pairs the compass construction of a square (\(s = \sqrt{2}R\)) and an equilateral triangle (\(s = \sqrt{3}R\)) inscribed in a circle of radius \(R\); Folios 2–3 solve for the Cartesian intersection \(N = (X_N, Y_N)\) of two diagonal/edge cevians inside a square of side \(s\); Folio 4–5 analyze subdivided equilateral triangle lattices of side 6 and side 1; and Folio 6 summarizes Carnot’s Theorem (\(OM_A + OM_B + OM_C = R + r\)) as the foundation of the classic Japanese Sangaku theorem on cyclic polygon triangulations.

Key Algebraic Identities

\[s_{\square} = \sqrt{2}\,R, \qquad s_{\triangle} = \sqrt{3}\,R, \qquad OM_A + OM_B + OM_C = R + r\]

\[X_N = \frac{s \cdot u v}{u + v - u v}, \qquad Y_N = \frac{s \cdot u (1 - v)}{u + v - u v}\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.45 Squares, Subdivided Lattices, and the Japanese Theorem — Folios (Folios 0287–0292, 6 scanned manuscript pages in resources/seelig/45_squares)
Euclid’s Elements	IV.6, IV.7 (EuclidElements)
GEOMETOR Elements (G-Index)	G.9 figure
Encyclopedia of Triangle Centers	General projective / metric locus (Kimberling1998)
Seelig Spreadsheets	Polygon Table, General Reference (Spreadsheets)

Constructive Dependency Graph

Prerequisites: F.4 Carnot’s Theorem, F.18 Equilateral Triangle, F.40 Regular Polygons and Nested Radicals