W.14 Menelaus, Ceva & The Alternate Spreads Theorem

Multiplicative Quadrance and Spread Balance for Collinearity and Concurrency

If either a transversal line l cuts the sidelines A_2 A_3, A_3 A_1, A_1 A_2 of a non-null triangle at B_1, B_2, B_3 (Menelaus) or three cevians through a point A_0 meet the opposite sidelines at B_1, B_2, B_3 (Ceva), then the six segment quadrances satisfy Q(A_2, B_1) * Q(A_3, B_2) * Q(A_1, B_3) = Q(B_1, A_3) * Q(B_2, A_1) * Q(B_3, A_2).

Interactive GEOMETOR construction for W.14 (Menelaus, Ceva & The Alternate Spreads Theorem). Hover any geometric label in the text or figure to highlight the corresponding element.

A_1 : Point A_2 : Point A_3 : Point A_0 : Point ===== t = polygon(A_1, A_2, A_3) B_1 = meet(line(A_2, A_3), line(A_1, A_0)) B_2 = meet(line(A_1, A_3), line(A_2, A_0)) B_3 = meet(line(A_1, A_2), line(A_3, A_0))

Mathematical Synthesis

Classically, Menelaus’ theorem (\(\prod = -1\)) and Giovanni Ceva’s theorem (\(\prod = +1\)) differ only in sign! Because quadrance squares signed ratios, both collinear transversals and concurrent cevians through A0 satisfy the identical necessary multiplicative law \(R_1 R_2 R_3 = P_1 P_2 P_3\) at division points B1, B2, B3 (Wildberger2005). (Consequently, the converse of neither quadrance product holds alone—their combined six-solution locus splits neatly into the collinear Menelaus configurations and concurrent Ceva configurations!)

Underpinning Ceva’s theorem is the Alternate Spreads Theorem (Theorem 79): for any point A0 in the plane of t, the alternating vertex spreads \(r_i = s(A_i A_{i+1}, A_i A_0)\) and \(p_i = s(A_i A_{i-1}, A_i A_0)\) satisfy \(r_1 r_2 r_3 = p_1 p_2 p_3\)—an invariant that generalizes to any \(n\)-gon and drives the algebraic solution of Hansen's problem (W.25).

Key Algebraic Identities

\[Q(A_2, B_1)\, Q(A_3, B_2)\, Q(A_1, B_3) = Q(B_1, A_3)\, Q(B_2, A_1)\, Q(B_3, A_2)\]

\[\text{Alternate Spreads:}\quad s(A_1 A_2, A_1 A_0)\, s(A_2 A_3, A_2 A_0)\, s(A_3 A_1, A_3 A_0) = s(A_1 A_3, A_1 A_0)\, s(A_2 A_1, A_2 A_0)\, s(A_3 A_2, A_3 A_0)\]

Symbolic SymPy Verification

Invariant / Theorem Check	Exact SymPy Value	Verification Status
Cevian feet B_1, B_2, B_3	B_1=[180/23, 108/23], B_2=[48/25, 108/25], B_3=[11/2, 0]	Exact rational intersections
Ceva Quadrance Balance LHS == RHS	2070829332/66125 == 2070829332/66125	Verified exact equality
Alternate Spreads Product r_1*r_2*r_3 == p_1*p_2*p_3	14907321/1026238660 == 14907321/1026238660	Theorem 79 verified

Cross-Repository Canon

Executable Demonstration	W.14 Menelaus, Ceva & The Alternate Spreads Theorem — Code (Standalone model.py & w-14.svg)
Thematic Part	Part III: Universal Triangle, Conic & Chromogeometry
Euclid’s Elements	VI.2 (EuclidElements)
GEOMETOR Elements	G.5 surface
GEOMETOR Fundamentals	F.7 Ceva and Menelaus (26 manuscript folios, SeeligNotebooks)
Sister Codices	geometor/pappus (projective incidence & transversal invariants)

Constructive Dependency Graph

Depends on: W.01 Planar Points, Line Proportions & Incidence Determinants, W.08 Law III: The Spread Law & Extended Circumquadrance Law

Required for: W.25 Snellius–Pothenot Resection & Hansen’s Surveying Problem