Unit III — Scholarly & GEOMETOR Symbolic Commentary

Exact SymPy Verification of Planar Menelaus, Gnomonic Projection & Wildberger Quadrances

1. Exact Symbolic Verification of Planar Menelaus & Its Universal Converse

In construct_menelaus_planar(), triangle \(A = (3, 6)\), \(B = (0, 0)\), \(C = (9, 0)\) is cut by transversal \(DF\) through \(D = (12, 0)\) (external on \(BC\)) and \(F = (2, 4)\) (internal on \(AB\)), meeting \(CA\) at \(E = (7, 2)\) with auxiliary parallel \(CH \parallel DF\) meeting \(AB\) at \(H = (3/2, 3)\):

\[\frac{\overline{AF}}{\overline{FB}} = \frac{1}{2}, \qquad \frac{\overline{BD}}{\overline{DC}} = -4, \qquad \frac{\overline{CE}}{\overline{EA}} = \frac{1}{2} \quad\Longrightarrow\quad \left(\frac{1}{2}\right)(-4)\left(\frac{1}{2}\right) = -1.\]

Furthermore, verify_planar_menelaus_universal() proves the universal algebraic identity over \(\mathbb{Q}[x_A, y_A, x_B, y_B, x_C, y_C, u, v, w]\): for \(F = (1-u)A + uB\), \(D = (1-v)B + vC\), \(E = (1-w)C + wA\),

\[\begin{split}\det\begin{pmatrix} x_D & y_D & 1 \\ x_E & y_E & 1 \\ x_F & y_F & 1 \end{pmatrix} \;\equiv\; \det\begin{pmatrix} x_A & y_A & 1 \\ x_B & y_B & 1 \\ x_C & y_C & 1 \end{pmatrix} \cdot \Bigl(u v w + (1 - u)(1 - v)(1 - w)\Bigr),\end{split}\]

making both the Forward Theorem and the Converse Theorem two halves of a single polynomial factorization!

2. Rational Trigonometry Reformulation (Wildberger Quadrances & Spreads)

A = (4, 8) B = (0, 0) C = (10, 0) D = (15, 0) F = (2, 4) E = (17/2, 2) H = (4/3, 8/3) A B C D F E H
Rational Trigonometry view of Planar Menelaus: polynomial quadrance balance Q(A,F)·Q(B,D)·Q(C,E) = Q(F,B)·Q(D,C)·Q(E,A) and Spread Law telescoping across △ABC and transversal DF.

In Norman J. Wildberger’s Rational Trigonometry (Divine Proportions, 2005, cf. geometor/wildberger and menelaus_rational_trig_invariants()), replacing Euclidean distance and angle with quadrance and spread \(Q(P, R) = (x_R - x_P)^2 + (y_R - y_P)^2\) and \(s(\ell_1, \ell_2)\) eliminates all square roots and transcendental functions across △ABC and transversal DF. On our canonical configuration \(A = (3,6), B = (0,0), C = (9,0)\) with \(D = (12,0), E = (7,2), F = (2,4)\):

  • \(Q(A,F) = 5\), \(Q(F,B) = 20\), \(Q(B,D) = 144\), \(Q(D,C) = 9\), \(Q(C,E) = 8\), \(Q(E,A) = 32\),

  • \(5 \times 144 \times 8 = 5760 = 20 \times 9 \times 32\), and the Spread Law across the three sub-triangles \(\triangle AEF\), \(\triangle BDF\), \(\triangle CDE\) telescopes identically to \(1\)!

3. Exact 3D Unit-Sphere & Gnomonic Verification (geometor.menelaus.spherical)

Notice how Menelaus’s chordal proof ([§3.1], Halley p. 80) and Ptolemy’s proof ([§3.5], Heiberg pp. 74–75) operate by intersecting rays from the sphere’s center \(O = (0,0,0)\) with an affine plane—which is central (gnomonic) projection \(\Pi_z(x, y, z) = (x/z, y/z)\) onto the tangent plane \(z = 1\)! In spherical_menelaus_default() and gnomonic_projection_menelaus():

  • Vertices \(A = (0,0,1)\), \(B = (3/5, 4/5, 0)\), \(C = (2/3, 2/3, 1/3) \in S^2(\mathbb{Q})\) cut by transversal plane \(2x - 3y + z = 0\) meet in radical traces \(D \in \mathbb{Q}(\sqrt{181})\), \(E = (\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}})\), \(F \in \mathbb{Q}(\sqrt{61})\).

  • Although the six arc sines lie in quadratic extensions (\(\sin\widehat{AF} = \frac{5}{\sqrt{61}}\), \(\sin\widehat{FB} = \frac{6}{\sqrt{61}}\), \(\sin\widehat{BD} = \frac{6}{\sqrt{181}}\), \(\sin\widehat{DC} = -\frac{5}{3\sqrt{181}}\), \(\sin\widehat{CE} = \frac{\sqrt{6}}{9}\), \(\sin\widehat{EA} = \frac{\sqrt{6}}{3}\)), every quadratic radical cancels pairwise via \(\frac{[\mathbf{P}, \mathbf{Q}, \mathbf{n}_t]}{[\mathbf{Q}, \mathbf{R}, \mathbf{n}_t]}\):

\[\left(\frac{5}{6}\right) \cdot \left(-\frac{18}{5}\right) \cdot \left(\frac{1}{3}\right) = -1, \qquad \frac{\operatorname{crd}(2\widehat{AF})}{\operatorname{crd}(2\widehat{FB})} \cdot \frac{\operatorname{crd}(2\widehat{BD})}{\operatorname{crd}(2\widehat{DC})} \cdot \frac{\operatorname{crd}(2\widehat{CE})}{\operatorname{crd}(2\widehat{EA})} = 1.\]