GEOMETOR feuerbach

Multi-Treatise Bilingual Book Codex & Exact Symbolic Triangle-Center Architecture

This Digital Scriptorium edition unites in a single computational and philological codex the two foundational treatises that transformed classical Euclidean triangle geometry into a unified science of collinear centers, symmetric polynomial invariants, and circle tangencies:

1. Leonhard Euler — Solutio facilis problematum quorumdam geometricorum difficillimorum (Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae, Tom. XI pro Anno 1765, publ. 1767, pp. 103–123, Eneström E325): verbatim 1767 St. Petersburg Latin alongside an original GEOMETOR English translation, deriving the six mutual quadrances among the orthocenter \(E = X(4)\), centroid \(F = X(2)\), incenter \(G = X(1)\), and circumcenter \(H = X(3)\), discovering the Euler Line, and reconstructing the triangle via the side-length cubic \(z^3 - pz^2 + qz - r = 0\).

2. Karl Wilhelm Feuerbach — Eigenschaften einiger merkwürdigen Punkte des geradlinigen Dreiecks (Nürnberg: Riegel und Wießner, 1822, pp. I–XVI, 1–62, with Karl Buzengeiger’s Vorrede): verbatim 1822 German alongside an original GEOMETOR English translation, introducing the orthic inradius \(\rho = 2R\cos\alpha\cos\beta\cos\gamma\), establishing the Nine-Point Circle centered at \(L = X(5)\), proving in § 57 its internal tangency to the incircle at the Feuerbach Point \(X(11)\) and external tangency to all three excircles, and concluding with a pure Euclidean Book I/III/VI synthetic Anhang (1.–9. Satz).

Every algebraic identity across both treatises is verified to exact symbolic zero in sympy (prove_euler_1765_identities()), and every geometric center and tangency locus is constructed without floating-point numbers on the canonical Heronian (15, 13, 14) triangle A = (0, 0), B = (14, 0), C = (5, 12) (construct_feuerbach_tangency()), where X₅ = (6, 63/16) and X₁₁ = (6, 8). For cevian concurrency and projective transversal duality, see the companion codices Ceva (1678) and Menelaus (Sphaerica).

Primary Treatises (4-Track Bilingual Codices)

• SOLVTIO FACILIS PROBLEMATVM QVORVMDAM GEOMETRICORVM DIFFICILLIMORVM (1765/1767)
  • Conspectus Operis (4 Units × 4 Synchronized Tracks)
• EIGENSCHAFTEN EINIGER MERKWÜRDIGEN PUNKTE DES GERADLINIGEN DREIECKS UND MEHRERER LINIEN (1822)
  • Conspectus Operis (6 Units × 4 Synchronized Tracks)

Scholarly Apparatus

• Cross-Treatise Concordance
  • Notation Rosetta Stone
  • Theorem & Proposition Concordance
• Bibliography & Cited Sources
  • Leonhard Euler — Solutio facilis problematum quorumdam geometricorum difficillimorum (1765/1767)
  • Karl Wilhelm Feuerbach — Eigenschaften einiger merkwürdigen Punkte des geradlinigen Dreiecks (1822)
  • Giovanni Ceva — De lineis rectis se invicem secantibus statica constructio (1678)
  • Menelaus of Alexandria — Sphaerica (c. 100 CE)
  • Euclid of Alexandria — Elements (Stoicheia)
  • Euclid of Alexandria — Data
  • Simon Antoine Jean L’Huilier — Analogies entre les triangles plans et sphériques (1810)
  • Giovanni Francesco Fagnano — Problemata quaedam ad methodum maximorum et minimorum spectantia (1775)
  • Nicolaus Fuß — De quadrilateris quibus circulum tam inscribere quam circumscribere licet (1797)
  • Charles Julien Brianchon & Jean-Victor Poncelet — Recherches sur la détermination d’une hyperbole équilatère (1821)
  • Olry Terquem — Considérations sur le triangle rectiligne (1842)
  • Nathan Altshiller-Court — College Geometry (1952)
  • Clark Kimberling — Encyclopedia of Triangle Centers (ETC)
  • Norman J. Wildberger — Divine Proportions: Rational Trigonometry to Universal Geometry (2005)
• Biographical Register of Persons
  • Leonhard Euler (1707 – 1783)
  • Karl Wilhelm Feuerbach (1800 – 1834)
  • Karl Heribert Ignaz Buzengeiger (1771 – 1835)
  • Euclid of Alexandria (fl. c. 300 BCE)
  • Apollonius of Perga (c. 240 – c. 190 BCE)
  • Pappus of Alexandria (c. 290 – c. 350 CE)
  • Menelaus of Alexandria (c. 70 – 140 CE)
  • Giovanni Ceva (1647 – 1734)
  • Matthew Stewart (1717 – 1785)
  • Giovanni Francesco Fagnano dei Toschi (1715 – 1797)
  • Simon Antoine Jean L’Huilier (1750 – 1840)
  • Nicolaus Fuß (1755 – 1826)
  • Charles Julien Brianchon (1783 – 1864)
  • Jean-Victor Poncelet (1788 – 1867)
  • Olry Terquem (1782 – 1862)
  • Nathan Altshiller-Court (1881 – 1968)
  • Clark Kimberling (b. 1942)
  • Norman J. Wildberger (b. 1956)
• Technical Glossary
  • Euler Line (Linea Euleri / Eulersche Gerade)
  • Nine-Point Circle (Kreis durch die Fußpunkte der Perpendikel)
  • Feuerbach Point X(11) (Innerer Berührungspunkt)
  • Tritangent Circles (Vier berührende Kreise)
  • Orthic Triangle (Fußpunktdreieck \(\triangle MNP\))
  • Feuerbach Triangle (\(\triangle Fe_a Fe_b Fe_c\))
  • Quadrance
  • Harmonic Range (\((X_4, X_2; X_5, X_3) = -1\))
  • Casus Irreducibilis
  • Elementary Symmetric Polynomials \((p, q, r)\)