F.47 The UVW Coordinate Triangle

Altitude-Origin Cartesian Frame A(-U,0), B(W,0), C(0,V) and 2x2 Cevian Trace Matrices

Placing the altitude foot H_C at the origin (0, 0) with A = (-U, 0), B = (W, 0), and C = (0, V) parameterizes any triangle by three numbers (U, V, W) with a = sqrt(V^2 + W^2), b = sqrt(U^2 + V^2), c = U + W, and reduces the cevian traces D, E, F through P(p, q) to explicit 2x2 rational matrix formulas.

Interactive GEOMETOR construction for F.47 (The UVW Coordinate Triangle). Hover any geometric label in the text or figure to highlight the corresponding element.

H_C : Point A : Point B : Point C : Point P : Point ===== s_ab = segment(A, B) s_ch = segment(C, H_C) D = intersection(line(B, C), line(A, P)) E = intersection(line(C, A), line(B, P)) F = intersection(line(A, B), line(C, P))

Mathematical Synthesis

Across 11 folios (SeeligNotebooks), Seelig develops his signature coordinate apparatus: The UVW Triangle. By placing the altitude foot HC at the Cartesian origin \((0,0)\) so that A \(= (-U, 0)\), B \(= (W, 0)\), and C \(= (0, V)\), every side and line equation enters double-intercept form (\(AC: -x/U + y/V = 1\) and \(BC: x/W + y/V = 1\)). When \((U, V)\) and \((V, W)\) are chosen as legs of two Pythagorean triples sharing height \(V\)—such as \((U,V,W) = (5,12,16)\) on Folios 3–4—the juxtaposed triangle has integer sides \((a,b,c) = (20, 13, 21)\) and integer area \(K = 126\) (a Heronian triangle).

Given an interior perspector P \(= (p, q)\), Seelig formulates \(2\times 2\) matrix systems for the cevian traces D \(= BC \cap AP\), E \(= CA \cap BP\), and F \(= AB \cap CP\), testing the formulas on \((U,V,W,p,q) = (8,8,16,4,4)\) to obtain exact rational traces \(D = (6.4, 4.8), E = (-2, 6), F = (8, 0)\) and verifying Ceva’s ratio product \(1\).

Key Algebraic Identities

\[A = (-U, 0), \qquad B = (W, 0), \qquad C = (0, V), \qquad a^2 = V^2 + W^2,\; b^2 = U^2 + V^2,\; c = U + W\]

\[F_x = \frac{p V}{V - q}, \qquad D = \left(\frac{W V (U + p)}{V(U + W) + q W},\; \frac{V q (U + W)}{V(U + W) + q W}\right)\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.47 The UVW Coordinate Triangle — Folios (Folios 0295–0305, 11 scanned manuscript pages in resources/seelig/47_u-v-w)
Euclid’s Elements	I.47, VI.2 (EuclidElements)
GEOMETOR Elements (G-Index)	G.4 segment, G.5 surface
Encyclopedia of Triangle Centers	X(2) — Centroid, X(4) — Orthocenter (Kimberling1998)
Seelig Spreadsheets	UVW Triangle, Ceva & Menelaus (Spreadsheets)

Constructive Dependency Graph

Prerequisites: F.7 Ceva and Menelaus, F.35 Obtuse and Acute Triangles

Required for: F.54 HP-35s Triangle Program and Coordinate Scripts