F.44 Homothetic Ratios and Crossed Diagonals

Internal and External Division of Parallel Uprights from a Center of Similitude

Given two parallel uprights of heights h_1 < h_2 separated by horizontal distance d and viewed from their external center of similitude O, the crossed diagonals meet at horizontal distance x = d h_1 / (h_1 + h_2) from the first upright, while O lies at distance y - x = d h_1 / (h_2 - h_1) to its left, forming a harmonic range.

Interactive GEOMETOR construction for F.44 (Homothetic Ratios and Crossed Diagonals). Hover any geometric label in the text or figure to highlight the corresponding element.

A : Point B : Point P_1 : Point P_2 : Point ===== s_1 = segment(A, P_1) s_2 = segment(B, P_2) O_ext = intersection(line(A, B), line(P_1, P_2)) O_int = intersection(line(A, P_2), line(B, P_1))

Mathematical Synthesis

Across five folios (SeeligNotebooks), Seelig analyzes the projective and metric proportions of two homothetic right triangles sharing an apex angle \(\phi\) at Oext with parallel vertical uprights \(h_1 < h_2\) separated by distance \(d\) along the baseline A B. By similar triangles (Euclid Elements VI.2 and VI.4), the crossed diagonals \(AP_2\) and \(BP_1\) intersect at Oint at horizontal distance \(x = d\frac{h_1}{h_1+h_2}\) from A (and height \(h = \frac{h_1 h_2}{h_1+h_2}\), half the harmonic mean), while the external apex Oext lies at distance \(y - x = d\frac{h_1}{h_2-h_1}\) from A.

Adding the internal and external distances yields the total span between the two centers of similitude \(y = \frac{2 d h_1 h_2}{h_2^2 - h_1^2}\) and its midpoint \(z = \frac{d h_1 h_2}{h_2^2 - h_1^2}\), reproducing the Apollonian harmonic conjugate formulas with \(k = h_2/h_1\).

Key Algebraic Identities

\[x = d\frac{h_1}{h_1 + h_2}, \qquad y - x = d\frac{h_1}{h_2 - h_1}, \qquad h = \frac{h_1 h_2}{h_1 + h_2}\]

\[y = \frac{2 d h_1 h_2}{h_2^2 - h_1^2}, \qquad z = \frac{y}{2} = \frac{d h_1 h_2}{h_2^2 - h_1^2}\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.44 Homothetic Ratios and Crossed Diagonals — Folios (Folios 0282–0286, 5 scanned manuscript pages in resources/seelig/44_ratios)
Euclid’s Elements	VI.2, VI.4 (EuclidElements)
GEOMETOR Elements (G-Index)	G.5 surface
Encyclopedia of Triangle Centers	General projective / metric locus (Kimberling1998)
Seelig Spreadsheets	Centers of Similitude, Harmonic Conjugates (Spreadsheets)

Constructive Dependency Graph

Prerequisites: F.5 Centers of Similitude, F.25 Harmonic Conjugates, F.27 Harmonic Mean