Unit 3 Summary: Ceva, Menelaus, Similitude & Anharmonic Projectivity¶

Mathematical Synthesis of Casey’s Book VI, Sections I–III & VI

1. Directed Ratios & Projective Duality Inside Euclidean Geometry¶

Unit 3 marks the transition in Casey’s Sequel from metric Euclidean geometry to projective invariants:

Ceva vs. Menelaus Duality: Both theorems evaluate the identical cyclic product $\frac{BD}{DC}\cdot\frac{CE}{EA}\cdot\frac{AF}{FB}$ on $\triangle ABC$: $+1$ characterizes concurrency of three lines (Ceva), while $-1$ characterizes collinearity of three points (Menelaus).
Homothecy & Monge’s Theorem: Two circles possess external and internal centres of similitude $S_e, S_i$ dividing $O_1 O_2$ harmonically in ratio $\pm r_1 : r_2$; Menelaus’s theorem immediately forces the six centres of similitude of three circles onto Monge’s four axes of similitude.
Complete Quadrilateral Harmonic Invariance: Pure linear incidence alone—without compass or metric measurement—constructs the fourth harmonic conjugate of three collinear points via the third diagonal $[EF]$ of a complete quadrilateral (0117.g1.svg).
Anharmonic Pencils on a Circle: Inscribed angle equality (Euclid III.21) makes the cross-ratio $P(ABCD)$ invariant on a circle, unlocking short synthetic proofs of Desargues’s Homology Theorem and Pascal’s Hexagrammum Mysticum.

Exact Rational Coordinates & Cross-Ratios of the Complete Quadrilateral¶
Projective Element	Incidence Definition	Exact Rational Value
Quadrangle Base Vertices	$A, B, C, D$	$(0, 0), \; (6, 0), \; (4, 2), \; (2, 3)$
Opposite Side Intersections	$E = [AB]\cap[CD], \; F = [AD]\cap[BC]$	$E = (8, 0), \; F = \left(\frac{12}{5}, \frac{18}{5}\right)$
Internal Diagonal Centre	$P = [AC]\cap[BD]$	$P = \left(\frac{18}{5}, \frac{9}{5}\right)$
Third Diagonal Harmonic Meets	$Q_a = [AC]\cap[EF], \; Q_b = [BD]\cap[EF]$	$Q_a = \left(\frac{9}{2}, \frac{9}{4}\right), \; Q_b = (-6, 9)$
Verified Cross-Ratios	$(A,C;P,Q_a), \; (B,D;P,Q_b), \; (E,F;Q_a,Q_b)$	$-1, \; -1, \; -1$