F.30 Isosceles Triangles

Perimeter-Area Cubic, Cardano’s Depressed Cubic, and Newton’s Right-Triangle Formula

An isosceles triangle of given semiperimeter s and area K has half-base x = c/2 governed by the cubic 2 s x^3 - s^2 x^2 + K^2 = 0, which under x = y + s/6 reduces to Cardano’s depressed cubic y^3 - (s^2/12) y - (s^3/54 - K^2/(2s)) = 0.

Interactive GEOMETOR construction for F.30 (Isosceles Triangles). Hover any geometric label in the text or figure to highlight the corresponding element.

A : Point B : Point M : Point ===== c_base = segment(A, B) perp = perpendicular(M, c_base) C = point_on(perp) a_leg = segment(B, C) b_leg = segment(C, A)

Mathematical Synthesis

Citing Baloglou and Helfgott (Forum Geometricorum 8, 2008, pp. 13–25, BaloglouHelfgott2008), Peterson’s mathematical blog, and Newton’s Universal Arithmetick (NewtonArithmetica), Seelig investigates how prescribing the semiperimeter \(s\) and area \(K\) determines a triangle (SeeligNotebooks). Whereas a right triangle has a linear hypotenuse formula \(c = s - K/s\) (Newton), an isosceles triangle with legs \(a = b = s - x\) and half-base \(x = c/2\) satisfies \(h = \sqrt{s(s-2x)}\) and \(K = x\sqrt{s(s-2x)}\), leading to the cubic \(2sx^3 - s^2 x^2 + K^2 = 0\).

Applying the Tschirnhaus shift \(x = y + s/6\) (with \(a = -s/2, b = 0, c = K^2/(2s)\)), Seelig reduces the equation to Cardano’s depressed cubic \(y^3 + py + q = 0\) with \(p = -s^2/12\) and \(q = -s^3/54 + K^2/(2s)\), analyzing the discriminant threshold \(K_{\max} = s^2/(3\sqrt{3})\) achieved at the equilateral triangle (\(x = s/3\)).

Key Algebraic Identities

\[a = b = s - x, \qquad x = \frac{c}{2}, \qquad K = x\sqrt{s(s - 2x)} \implies 2 s x^3 - s^2 x^2 + K^2 = 0\]

\[x = y + \frac{s}{6} \implies y^3 - \frac{s^2}{12} y + \left(\frac{K^2}{2s} - \frac{s^3}{54}\right) = 0, \qquad c_{\text{right}} = s - \frac{K}{s}\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.30 Isosceles Triangles — Folios (Folios 0210–0214, 5 scanned manuscript pages in resources/seelig/30_isosceles)
Euclid’s Elements	I.5, I.6 (EuclidElements)
GEOMETOR Elements (G-Index)	G.1 point
Encyclopedia of Triangle Centers	General projective / metric locus (Kimberling1998)
Seelig Spreadsheets	Isosceles Triangle, Right Triangle (Spreadsheets)

Constructive Dependency Graph

Prerequisites: F.3 Biquadratic Forms in Heron’s Formula