F.33 Euclidean (m, n) Generators

Parametric Pythagorean Triple Table for 2 <= m <= 7

For integers m > n >= 1, Euclid’s quadratic generators a = m^2 - n^2, b = 2 m n, and c = m^2 + n^2 satisfy a^2 + b^2 = c^2 and inradius r = n(m - n).

Interactive GEOMETOR construction for F.33 (Euclidean (m, n) Generators). Hover any geometric label in the text or figure to highlight the corresponding element.

A : Point B : Point C : Point ===== a = segment(B, C) b = segment(C, A) c = segment(A, B)

Mathematical Synthesis

This single-folio reference table (SeeligNotebooks) tabulates Euclid’s parametric generators \((m, n, m^2-n^2, 2mn, m^2+n^2)\) (from Elements Book X, Lemma 1 to X.29) across 11 integer pairs for \(2 \le m \le 7\): from \((m,n) = (2,1) \mapsto (3,4,5)\) through \((3,2)\mapsto(5,12,13)\), \((4,1)\mapsto(15,8,17)\), \((4,3)\mapsto(7,24,25)\), \((5,2)\mapsto(21,20,29)\), \((5,4)\mapsto(9,40,41)\), \((6,1)\mapsto(35,12,37)\), \((6,5)\mapsto(11,60,61)\), \((7,2)\mapsto(45,28,53)\), \((7,4)\mapsto(33,56,65)\), to \((7,6)\mapsto(13,84,85)\).

Every row in Seelig’s table omits common-factor pairs (such as \((3,1)\) or \((4,2)\)) so that all 11 generated triples are primitive Pythagorean triple right triangles with integer inradius \(r = n(m-n)\) and semiperimeter \(s = m(m+n)\).

Key Algebraic Identities

\[a = m^2 - n^2, \qquad b = 2 m n, \qquad c = m^2 + n^2, \qquad a^2 + b^2 = (m^2 + n^2)^2 = c^2\]

\[s = m(m + n), \qquad r = n(m - n), \qquad K = m n (m^2 - n^2)\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.33 Euclidean (m, n) Generators — Folios (Folios 0223–0223, 1 scanned manuscript page in resources/seelig/33_m-n)
Euclid’s Elements	I.47, X.29 (EuclidElements)
GEOMETOR Elements (G-Index)	G.11 rectilineal figures
Encyclopedia of Triangle Centers	General projective / metric locus (Kimberling1998)
Seelig Spreadsheets	Pythagorean Triples, Right Triangle (Spreadsheets)

Constructive Dependency Graph

Prerequisites: F.29 Inradius

Required for: F.42 Altitude of a Pythagorean Triple Triangle