Chapter VI: Transversals — Commentary

Constructive Analysis, Historical Provenance & GEOMETOR Notes (0170–0184, pp. 151–165)

Mathematical Architecture & Signed Ratios

Chapter VI (§§ 306–344, pp. 151–165) introduces directed segments (\(AB = -BA\), Chasles / Möbius) to unify collinearity and concurrency:

• Introductory & Stewart’s Theorem (§§ 306–309): Euler's four-point identity \(AB \cdot CD + AC \cdot DB + AD \cdot BC = 0\) (§ 307) and Stewart's cevian metric identity (§ 308).

• Menelaus’ Theorem (§§ 310–325): Signed transversal product \((AF/FB)(BD/DC)(CE/EA) = -1\) (Menelaus of Alexandria), applied immediately to collinearity of external angle-bisector traces, Pascal's mystic hexagram, and centers of similitude of three circles (Monge).

• Ceva’s Theorem (§§ 326–344): Dual concurrency criterion \((AF/FB)(BD/DC)(CE/EA) = +1\) (Ceva 1678), establishing concurrency of medians, altitudes, bisectors, the Gergonne point \(X(7)\) (§ 331, Gergonne), the Nagel point \(X(8)\) (§ 333, Nagel), isotomic conjugates (§ 334), and Desargues' Two-Triangle Theorem (§ 343).

Cross-Repository GEOMETOR Links

• geometor/pappus

• geometor/etc (X(7) Gergonne, X(8) Nagel, X(10) Spieker, Isotomic Conjugates)