F.3 Biquadratic Forms in Heron’s Formula

Symmetric Quadratic Substitutions, Circumcircle Families, and Extremal Side Lengths

Under the linear substitution X = a/2, W = (b + c)/2, and Y = (c - b)/2, Heron’s area formula reduces to the biquadratic product K = sqrt((W^2 - X^2)(X^2 - Y^2)).

Interactive GEOMETOR construction for F.3 (Biquadratic Forms in Heron’s Formula). 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) H_C = foot(C, c) h_c = segment(C, H_C)

Mathematical Synthesis

In Topic 03 (SeeligNotebooks), Seelig reformulates Heron's formula \(K = \sqrt{s(s-a)(s-b)(s-c)}\) by introducing the symmetric half-sums and half-differences \(X = a/2\), \(W = (b+c)/2\), and \(Y = (c-b)/2 = -Z\). Because \(s = W + X\), \(s - a = W - X\), \(s - b = X + Y\), and \(s - c = X - Y\), the squared area collapses immediately into the biquadratic difference of squares \(K^2 = (W^2 - X^2)(X^2 - Y^2)\).

Seelig then studies a triangle inscribed in a circle of constant circumradius \(R\) with fixed chord \(c\) and varying side \(a\), expressing \(b(a; c, R)\) via the Law of Sines and solving the stationarity condition via polynomial division of \((x^3 - y^3)/(x - y)\) to obtain the extremal relation \({z_i^*}^2 = w_i^2 + w_i w_i' + {w_i'}^2\).

Key Algebraic Identities

\[X = \frac{a}{2}, \quad W = \frac{b + c}{2}, \quad Y = \frac{c - b}{2} \implies K = \sqrt{(W^2 - X^2)(X^2 - Y^2)}\]

\[16 K^2 = 2 a^2 b^2 + 2 b^2 c^2 + 2 c^2 a^2 - a^4 - b^4 - c^4, \qquad {z_i^*}^2 = w_i^2 + w_i w_i' + {w_i'}^2\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.3 Biquadratic Forms in Heron’s Formula — Folios (Folios 0011–0019, 9 scanned manuscript pages in resources/seelig/03_biquadratic-forms-in-herons-formula)
Euclid’s Elements	I.47, II.12, II.13 (EuclidElements)
GEOMETOR Elements (G-Index)	G.1 point, G.3 extremities of a segment
Encyclopedia of Triangle Centers	X(1) — Incenter, X(3) — Circumcenter, X(4) — Orthocenter (Kimberling1998)
Seelig Spreadsheets	General Reference, 5-6-7 A,B,C, Biquadratic Heron, Circumcircle Solver (Spreadsheets)

Constructive Dependency Graph

Required for: F.4 Carnot’s Theorem, F.6 Centroid, F.11 Circumcenter, F.13 Compare Triangles in Circles, F.14 Conway Notation, F.28 Incenter, F.30 Isosceles Triangles, F.35 Obtuse and Acute Triangles, F.50 Altitudes, Mitchell’s Reciprocal Heron Formula, and the Symphonic Theorem