F.52 Homogeneous Barycentric Coordinates

Classical Center Catalog X(1)–X(8), Sidelength Polynomials for X(5), and 5-6-7 Evaluations

Every triangle center X(n) has homogeneous barycentric coordinates (u : v : w) normalizable to absolute weights lambda_A + lambda_B + lambda_C = 1 such that X = lambda_A A + lambda_B B + lambda_C C; for the nine-point center N = X(5), lambda_A = 1/4 + (b^2 - c^2)(a^2 - (b^2 + c^2)) / (32 K^2).

Interactive GEOMETOR construction for F.52 (Homogeneous Barycentric Coordinates). Hover any geometric label in the text or figure to highlight the corresponding element.

A : Point B : Point C : Point ===== G = centroid(A, B, C) I = incenter(A, B, C) O = circumcenter(A, B, C) H = orthocenter(A, B, C) N = midpoint(O, H) euler = line(O, H)

Mathematical Synthesis

Spanning 15 folios (SeeligNotebooks), Seelig’s master treatise on homogeneous barycentric coordinates (HBC) bridges classical synthetic geometry, Kimberling’s Encyclopedia of Triangle Centers (KimberlingETC), and programmable calculator/spreadsheet computation. Folios 1–6 (bari-centers-001–006) tabulate the unnormalized homogeneous coordinates \((u:v:w)\), normalizing sums \(\Sigma = u+v+w\), and exact numerical evaluations on the canonical \((5,6,7)\) triangle for eight fundamental centers: incenter I \(= X(1) = (a:b:c)\), centroid G \(= X(2) = (1:1:1)\), circumcenter O \(= X(3) = (a^2 S_A : b^2 S_B : c^2 S_C)\), orthocenter H \(= X(4) = (S_B S_C : S_C S_A : S_A S_B)\), nine-point center N \(= X(5)\), symmedian point \(K = X(6) = (a^2:b^2:c^2)\), Gergonne point \(Ge = X(7)\), and Nagel point \(Na = X(8) = (s-a:s-b:s-c)\).

In Folios 7–15 (homogenus-barycentric-co001–009, dated FWS 2014/9/14), Seelig derives two explicit forms for the normalized barycentric coordinates of the nine-point center N (X(5) — Nine-Point Center): Form I (trigonometric, \(u = a\cos(\beta - \gamma)\)) and Form II (purely polynomial in the side lengths: \(\lambda_A = W + \frac{P^2/a^2 + a^4}{D}\) with \(D = 4U - 2S_4 = 16K^2\)), verifying Form II on the \((3,4,5)\) right triangle \((\frac{1}{4} : \frac{1}{4} : \frac{1}{2})\), the \((5,6,7)\) acute triangle \((0.28299 : 0.30208 : 0.41493)\), and a compass drawing of the \((7,8,9)\) nine-point circle (\(R_N = \frac{21}{20}\sqrt{5}\)).

Key Algebraic Identities

\[P = \lambda_A A + \lambda_B B + \lambda_C C \qquad (\lambda_A + \lambda_B + \lambda_C = 1), \qquad \lambda_A = \frac{u}{u + v + w} = \frac{[PBC]}{[ABC]}\]

\[X(1) = (a : b : c),\quad X(2) = (1 : 1 : 1),\quad X(3) = (a^2 S_A : b^2 S_B : c^2 S_C),\quad X(4) = (S_B S_C : S_C S_A : S_A S_B),\quad X(6) = (a^2 : b^2 : c^2)\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.52 Homogeneous Barycentric Coordinates — Folios (Folios 0331–0345, 15 scanned manuscript pages in resources/seelig/52_homogeneous-barycentric-coordinates)
Euclid’s Elements	IV.4, IV.5, VI.1 (EuclidElements)
GEOMETOR Elements (G-Index)	G.7 plane surface, G.8 boundary
Encyclopedia of Triangle Centers	X(1) — Incenter, X(2) — Centroid, X(3) — Circumcenter, X(4) — Orthocenter, X(5) — Nine-Point Center, X(6) — Symmedian (Lemoine) Point, X(7) — Gergonne Point, X(8) — Nagel Point (Kimberling1998)
Seelig Spreadsheets	Barycentric, Barycentric (2), 9-Point Circle, 5-6-7 A,B,C (Spreadsheets)

Constructive Dependency Graph

Prerequisites: F.6 Centroid, F.11 Circumcenter, F.14 Conway Notation, F.28 Incenter, F.34 Nine-Point Circle

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