EIGENSCHAFTEN EINIGER MERKWÜRDIGEN PUNKTE DES GERADLINIGEN DREIECKS UND MEHRERER LINIEN (1822)

Properties of Several Remarkable Points of the Rectilinear Triangle, and of Several Lines and Figures Determined by Them — Karl Wilhelm Feuerbach

Editio Princeps (Deutsch, 1822): Eigenschaften einiger merkwürdigen Punkte des geradlinigen Dreiecks, und mehrerer durch sie bestimmten Linien und Figuren. Eine analytisch-trigonometrische Abhandlung von Karl Wilhelm Feuerbach, der Philosophie Doktor. Mit einer Vorrede von Karl Buzengeiger, ordentlichem Professor der Mathematik an der großherz. badischen Universität zu Freiburg. Nürnberg, 1822. Bei Riegel und Wießner (pp. I–XVI, 1–62, Tab. I–II, Fig. 1–17).

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

Synchronized 4-Track Codex: English · Deutsch (1822) · Summary · Notes, paired with exact symbolic verification in geometor.feuerbach.euler and geometor.feuerbach.tangency.

„In jedem Dreieck trifft der Kreis, welcher durch die Fußpunkte seiner Perpendikel geht, zugleich die Seiten desselben in ihren Mitten. […] Der Kreis, welcher durch die Fußpunkte der Perpendikel eines Dreiecks geht, berührt alle vier die drei Seiten desselben berührenden Kreise, und zwar den innerhalb berührenden innerhalb, jeden der ausserhalb berührenden aber ausserhalb.“

“In every triangle, the circle which passes through the feet of its altitudes simultaneously meets the sides of the same at their midpoints. […] The circle which passes through the feet of the altitudes of a triangle touches all four circles touching the three sides of the same: namely, it touches the internally touching circle internally, whereas it touches each of the externally touching circles externally.”

—Karl Wilhelm Feuerbach, Eigenschaften, §§ 56–57 (p. 38)

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Conspectus Operis (6 Units × 4 Synchronized Tracks)

Unit 0 — Karl Buzengeiger’s Vorrede (pp. III–XVI, Freiburg, 16 March 1822)

English · Deutsch (1822) · Summary · Notes

Mathematics contemplated as Wissenschaft (Euclid’s Elements) versus Kunst (Greek geometric Analysis in Euclid's Data, Apollonius, Robert Simson’s restorations, and Diophantus), reflections on combinatorial analysis and factorials, and Buzengeiger’s commendation of his Freiburg pupil Feuerbach.

Unit I — Erster Abschnitt: The Four Tritangent Circle Centers (§§ 1–18, pp. 1–14)

English · Deutsch (1822) · Summary · Notes

Incenter \(S = X(1)\) and excenters \(S', S'', S'''\) as orthocentric system, L'Huilier’s symmetric law \(rr'r''r''' = \Delta^2\) (§ 2), \(r'r'' + r'r''' + r''r''' = s^2\) (§ 3), \(r' + r'' + r''' = r + 4R\) (§ 5), contact triangles \(\delta, \delta', \delta'', \delta'''\) (§ 8), and the six mutual center-chord laws \(A^2 + A'^2 = B^2 + B'^2 = C^2 + C'^2 = 16R^2\) (§§ 12–18).

Unit II — Zweiter Abschnitt: Orthocenter, Orthic Triangle & Inradius ρ (§§ 19–48, pp. 15–32)

English · Deutsch (1822) · Summary · Notes

Orthocenter \(O = X(4)\) as incenter of Fagnano’s minimal-perimeter orthic triangle \(\triangle MNP\) (§ 19), introduction of the orthic inradius \(\rho = 2R\cos\alpha\cos\beta\cos\gamma\) (§§ 23–25), orthic circumradius \(\tfrac{1}{2}R\) (§ 26), \(r^2 + r'^2 + r''^2 + r'''^2 = 4R(2R - \rho)\) (§ 28), and the equal circumradii \(R' = R'' = R''' = R\) of \(\triangle BCO, \triangle ACO, \triangle ABO\).

Unit III — Dritter & Vierter Abschnitt: The Nine-Point Circle & Feuerbach’s Tangency Theorem (§§ 49–60, pp. 33–41)

English · Deutsch (1822) · Summary · Notes

Complete verbatim edition of §§ 49–60: citation of Euler (1765) and Fuß (1797) for \(\overline{KS}^2 = R^2 - 2rR\) (§ 49), nine-point center \(L = X(5)\) bisecting \(KO\) (§§ 53–55), the nine concyclic points (§ 56), § 57 Feuerbach’s Theorem (\(LS = \tfrac{1}{2}R - r\) at \(X(11)\) and \(LS^{(k)} = \tfrac{1}{2}R + r^{(k)}\)), \(LS + LS' + LS'' + LS''' = 6R\) (§ 58), \(\sum LS_i^2 = R(13R - 4\rho)\) (§ 59), and the harmonic range \((O, G; L, K) = -1\) (§ 60), together with exact SymPy diagnosis of Feuerbach’s § 51 sign-substitution slip.

Unit IV — Fünfter Abschnitt: Proportional Contact Triangles & Excentral Orthocenter (§§ 61–87, pp. 41–55)

English · Deutsch (1822) · Summary · Notes

Geometric-mean area proportionals \(\mu^{(k)} : \delta^{(k)} = \delta^{(k)} : \Delta\) (§ 63), the five-triangle area law \(\mu + \mu' + \mu'' + \mu''' + \triangle MNP = 2\Delta\) (§ 64), half-perimeter law (§ 65), corner-circle inradii (§§ 71–73), and § 87 establishing that \(S = X(1)\) is the orthocenter of \(\triangle O'O''O'''\).

Unit V — Synthetischer Anhang: Pure Euclidean Proofs of Nine Core Propositions (1.–9. Satz, pp. 56–62)

English · Deutsch (1822) · Summary · Notes

Pure synthetic demonstrations grounded in Euclid's Elements Books I, III, and VI, culminating in 5. Satz (\(\triangle MNP : \triangle ABC = \rho : R\)), 6. Satz (\(CO \cdot OP = 2R\rho\)), 7. Satz (\(a^2 + b^2 + c^2 = 8R^2 + 4\rho R\)), and verbatim 9. Satz proving \(LM = LN = LP = \tfrac{1}{2}R\) for the midpoint \(L\) of \(OK\).

English Translation

• Unit 0: Preface (Vorrede) by Karl Buzengeiger (pp. III–XVI, March 1822)
• Unit I: First Section — On the Centers of the Four Tritangent Circles (§§ 1–18)
• Unit II: Second Section — The Orthocenter, Orthic Triangle & Inradius ρ (§§ 19–48)
• Unit III: The Nine-Point Circle, Feuerbach’s Theorem & The Euler Line (§§ 49–60, Complete)
• Unit IV: Fifth Section — Comparative Proportions Among Derived Triangles (§§ 61–87)
• Unit V: Synthetic Appendix (Anhang) — Pure Euclidean Proofs of Nine Theorems (pp. 56–62)