SOLVTIO FACILIS PROBLEMATVM QVORVMDAM GEOMETRICORVM DIFFICILLIMORVM (1765/1767)

An Easy Solution of Certain Very Difficult Geometric Problems — Leonhard Euler (Eneström E325)

Editio Princeps (Latina, 1767): Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae, Tom. XI pro Anno MDCCLXV, Petropoli: Typis Academiae Scientiarum, MDCCLXVII, pp. 103–123, Tab. II Fig. 1–9 (presented November 1763 / published 1767; Opera Omnia, Ser. I, Vol. 26, pp. 139–157).

English Translation: Original folio-verified translation curated for the GEOMETOR Digital Scriptorium.

Synchronized 4-Track Codex: English · Latina (1767) · Summary · Notes, paired with exact symbolic verification in geometor.feuerbach.euler.

„Tum vero omnibus difficultatibus feliciter superatis insignes quaedam affectiones inter illa quatuor puncta se prodent, quarum cognitio in Geometria haud leuis momenti est censenda.“

“Moreover, once all difficulties have been happily overcome, certain remarkable properties among those four points reveal themselves, the knowledge of which must be judged of no small importance in Geometry.”

—Leonhard Euler, Solutio facilis, § 3 (p. 104)

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Conspectus Operis (4 Units × 4 Synchronized Tracks)

Unit I — The Four Classical Triangle Centers & Orthogonal Coordinates (§§ 1–9, pp. 103–107)

English · Latina (1767) · Summary · Notes

Enumeration of the four classical concurrency centers \(E = X(4)\) (orthocenter), \(F = X(2)\) (centroid), \(G = X(1)\) (incenter), and \(H = X(3)\) (circumcenter), formulation of the inverse determination problem, and derivation of their rational orthogonal coordinates on base \(AB = c\).

Unit II — Symmetric Polynomials, Six Mutual Quadrances & The Euler Line (§§ 10–18, pp. 107–114)

English · Latina (1767) · Summary · Notes

Change of variables to the elementary symmetric polynomials \(p = a+b+c\), \(q = ab+ac+bc\), \(r = abc\), symmetric Heron expansion \(16A^2 = -p^4 + 4p^2q - 8pr\), closed-form derivation of all six squared inter-center distances (Formulas I–VI), discovery of the Euler Line (\(FH = \tfrac{1}{2}EF\), \(EH = \tfrac{3}{2}EF\)), and the compatibility law \(4GH^2 + 2EG^2 = 3EF^2 + 6FG^2\).

Unit III — The Inverse Reconstruction Problem & Side-Length Cubic (§§ 19–24, pp. 114–118)

English · Latina (1767) · Summary · Notes

Reduction of the inverse problem (“given the four centers \(E, F, G, H\), construct the triangle”) to auxiliary invariants \(P = p^2\), \(Q = r/p\), \(R = r^2/(4A^2)\), exact cancellation of the quadratic term \(Q^2\) yielding rational \((P, q, Q)\), Viète’s cubic \(z^3 - pz^2 + qz - r = 0\), and the integer benchmark \((5, 6, 7) \implies z^3 - 18z^2 + 107z - 210 = 0\).

Unit IV — Collinear Centers, Isosceles Degeneracy & Casus Irreducibilis (§§ 25–34, pp. 118–123)

English · Latina (1767) · Summary · Notes

Analysis of the degenerate case in which \(G\) lies on the Euler line \(EH\) (forcing an isosceles triangle with rational root \(z = c\)), the four positional regimes relative to midpoint \(I\), symmetric parameterization by \((e, f, k) = (EG, GH, EH)\), reality diorismos inequalities, and angle trisection in the casus irreducibilis.

English Translation

• Unit I: The Four Classical Triangle Centers & Orthogonal Coordinates (§§ 1–9)
• Unit II: Symmetric Polynomials, Six Mutual Quadrances & The Euler Line (§§ 10–18)
• Unit III: The Inverse Reconstruction Problem & Side-Length Cubic (§§ 19–24)
• Unit IV: Collinear Centers, Isosceles Degeneracy & Casus Irreducibilis (§§ 25–34)