← menelaus / GEOMETOR • menelaus

Symbolic Demonstrations¶

100% Exact SymPy & geometor.model Constructions of Planar, Spherical & Dual Menelaus Geometry

Every construction in geometor.menelaus executes with zero floating-point approximations over \(\mathbb{Q}\) and closed-form quadratic radical extensions, rendering interactive .euclid-graphic SVGs via scripts/build_figures.py. Hover or click any inline :g: geometric element reference below to highlight its exact counterpart inside the figure!

1. Planar Menelaus & Euclidean Parallel Auxiliary (Halley Lemma I / Almagest I.13)¶

Given reference triangle ABC (A = (3, 6), B = (0, 0), C = (9, 0)) and transversal DF cutting extended side BC externally at D ((12, 0)), side CA internally at E ((7, 2)), and side AB internally at F ((2, 4)), drawing auxiliary segment CH through C parallel to DF ([I.31]) meets AB at H ((3/2, 3)):

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

Planar Menelaus's Theorem: similar triangles via auxiliary segment CH parallel to transversal DF.

Constructed symbolically via construct_menelaus_planar().

2. Wildberger Rational Trigonometry Formulation (Quadrance & Spread Laws)¶

Using quadrance() and line_spread() (menelaus_rational_trig_invariants()), Planar Menelaus on ABC and transversal DF holds purely over \(\mathbb{Q}\) with zero square roots:

\[\frac{Q(A, F)\,Q(B, D)\,Q(C, E)}{Q(F, B)\,Q(D, C)\,Q(E, A)} = \frac{s(t, b)}{s(t, c)} \cdot \frac{s(t, c)}{s(t, a)} \cdot \frac{s(t, a)}{s(t, b)} = 1.\]

Rational Trigonometry formulation: spread ratios across sub-triangles △AFE, △BDF, and △CDE telescope identically to 1 over ℚ.

3. Converse of Menelaus Gallery (Four Projective Collinearity Theorems)¶

3.1 External Angle-Bisector Axis (Anti-Orthic Axis, cf. Sphaerica III.6)¶

On the scalene Heronian triangle ABC (A = (-9, 12), B = (0, 0), C = (7, 0), side lengths a = 7, b = 20, c = 15, integer incenter I = (1, 2)), construct_external_bisector_axis() constructs the three external angle bisectors meeting the opposite sides at De = (-21, 0), Ee = (21, -21/2), and Fe = (63/13, -84/13), whose signed ratio product \((-c/a)(-b/c)(-a/b) = (-15/7)(-20/15)(-7/20) = -1\) places De, Ee, and Fe on the anti-orthic axis DeEe!

External Angle-Bisector Axis (anti-orthic axis De–Ee–Fe) of Heronian △ABC with integer incenter I.

3.2 Desargues’s Two-Triangle Perspectivity Axis¶

In construct_desargues_axis(), applying Planar Menelaus three times to perspective ray-triangles \(\triangle OBC\), \(\triangle OCA\), and \(\triangle OAB\) from center O cancels the radial ratios along OA1, OB1, and OC1, proving that corresponding side intersections D, E, and F lie on Desargues’s axis DF.

Desargues's Two-Triangle Perspectivity Theorem proved via three applications of Planar Menelaus from center O.

3.3 Pascal’s Hexagrammum Mysticum¶

In construct_pascal_hexagram(), inscribing six rational vertices P1, P2, P3, P4, P5, P6 on \(x^2 + y^2 = 100\) and forming the diagonal triangle U, V, W of alternating sides, three applications of Planar Menelaus combined with Euclid III.35/III.36 power-of-a-point chord products prove that opposite-side intersections L, M, and N are collinear on Pascal’s Mystic Line LM.

Pascal's Hexagrammum Mysticum: opposite-side intersections L, M, and N align on Pascal's Mystic Line via triple Menelaus on △UVW.

3.4 Monge’s Three-Circle Similicenter Theorem¶

In construct_monge_exsimilicenters(), for circles centered at A, B, C with exact radii \(r_A = 1, r_B = 2, r_C = 4\), the three external centers of similitude Sbc, Sca, and Sab satisfy \((-r_A/r_B)(-r_B/r_C)(-r_C/r_A) = (-1/2)(-2/4)(-4/1) = -1\), placing them on Monge’s external axis SbcSca.

Monge's Three-Circle Theorem: external similicenters Sbc, Sca, and Sab align on Monge's axis via Converse of Menelaus.

4. Unit-Sphere Intrinsic Geometry, Spherical Menelaus & Cross-Ratio Pencil¶

4.1 Spherical Excess (\(A+B+C > \pi\)) & Polar Triangle AAA Congruence (Sphaerica I.11 & I.17)¶

Via polar_triangle(), spherical triangle A, B, C (AB, BC, CA) and its supplementary polar triangle A, B, C satisfy \(\cos a^* = -\cos A\), confirming that three dihedral angles uniquely determine all three arc lengths on \(S^2\) (Sphaerica Prop. I.17).

Intrinsic spherical triangle △ABC and its supplementary polar triangle △A*B*C* on the unit sphere S² (Sphaerica I.11 & I.17).

4.2 Spherical Menelaus Theorem (Sphaerica III.1) & Spherical Cross-Ratio Pencil (Sphaerica III.8)¶

On \(S^2(\mathbb{Q})\) with A = (0, 0, 1), B = (3/5, 4/5, 0), and C = (2/3, 2/3, 1/3) cut by transversal normal (2, -3, 1):

\[\frac{\sin_{\pm}\widehat{AF}}{\sin_{\pm}\widehat{FB}} \cdot \frac{\sin_{\pm}\widehat{BD}}{\sin_{\pm}\widehat{DC}} \cdot \frac{\sin_{\pm}\widehat{CE}}{\sin_{\pm}\widehat{EA}} = \left(\frac{5}{6}\right)\left(-\frac{18}{5}\right)\left(\frac{1}{3}\right) = -1, \qquad \tan\!\left(\frac{E}{2}\right) = \frac{1}{17} > 0.\]

In verify_spherical_cross_ratio_invariance(), four great circles concurrent at pole V cut oblique great circles X1X4 (X1, X2, X3, X4) and Y1Y4 (Y1, Y2, Y3, Y4) with identical spherical sine cross-ratio \(-2\):

Spherical pencil invariance (Sphaerica Prop. III.8): four great circles concurrent at pole V cut two transversals with identical sine cross-ratio −2.

5. Projective Point-Line Duality: Ceva (+1) vs. Menelaus (-1) & Trilinear Polars¶

Given perspector P = (3, 3) inside ABC, construct_trilinear_polar() constructs its concurrent cevians meeting the sides at Dc, Ec, Fc (Ceva product = +1). Intersecting the cevian sides with the extended sides of ABC produces the harmonic conjugates Dm, Em, Fm, satisfying \((B, C; D_c, D_m) = (C, A; E_c, E_m) = (A, B; F_c, F_m) = -1\), negating all three ratios \((-1)^3(+1) = -1\) and aligning them on the trilinear polar DmEm!

Ceva–Menelaus Harmonic Duality: harmonic conjugates Dm, Em, and Fm of cevian traces Dc, Ec, and Fc align on the trilinear polar of P.

Executable Demo Scripts¶

Run all three verification demos offline from the repository root:

/home/phi/PROJECTS/geometor/.venv/bin/python demos/planar_menelaus_demo.py
        /home/phi/PROJECTS/geometor/.venv/bin/python demos/spherical_menelaus_demo.py
        /home/phi/PROJECTS/geometor/.venv/bin/python demos/ceva_menelaus_duality_demo.py
        