Lines, Proportions & Spreads

A Line Is a Proportion Derived from the Relationship of Two Points

We are often taught to picture a line as an “infinite string of points.” Constructively, the truth is sharper:

A point on a line cannot be known unless that line intersects another element. Anything else is speculation.

When two distinct points \(P_1 = (x_1, y_1)\) and \(P_2 = (x_2, y_2)\) are given (G.2), aligning the unmarked straightedge projects a constant linear proportion [ P₁ P₂ ] across the field:

\[a x + b y + c = 0, \qquad a = y_1 - y_2, \quad b = x_2 - x_1, \quad c = x_1 y_2 - x_2 y_1\]

Notice the symmetry: interchanging \(P_1\) and \(P_2\) negates \((a, b, c)\) simultaneously, leaving the locus [ P₁ P₂ ] = [ P₂ P₁ ] invariant.

Exact Line–Line Selection

Two non-parallel lines \(a_1 x + b_1 y + c_1 = 0\) and \(a_2 x + b_2 y + c_2 = 0\) with non-vanishing determinant \(\Delta = a_1 b_2 - a_2 b_1 \neq 0\) select a unique rational point in the same algebraic number field as their coefficients:

\[x = \frac{b_1 c_2 - b_2 c_1}{a_1 b_2 - a_2 b_1}, \qquad y = \frac{c_1 a_2 - c_2 a_1}{a_1 b_2 - a_2 b_1}\]

Pure straightedge operations (projective incidence) never introduce square roots! Because of this algebraic closure, theorems of pure collinearity and concurrency—such as Pappus’s Hexagon Theorem and the Complete Quadrangle construction of the Harmonic Fourth (geometor/pappus)—hold over any commutative field.

Inclination Without Transcendental Angles: Spread

Rather than measuring the inclination of two lines with transcendental arccos angles, geometor.model and geometor/wildberger compute their exact rational spread:

\[s(\ell_1, \ell_2) = \frac{(a_1 b_2 - a_2 b_1)^2}{(a_1^2 + b_1^2)(a_2^2 + b_2^2)}\]

• \(s = 0\) — parallel or coincident lines (\(0^\circ\));

• \(s = \frac{1}{4}\) — \(30^\circ\) / \(150^\circ\) inclination;

• \(s = \frac{1}{2}\) — diagonal \(45^\circ\) inclination (\(\sqrt{2}\) square diagonal);

• \(s = \frac{3}{4}\) — equilateral \(60^\circ\) / \(120^\circ\) inclination (\(\sqrt{3}\) Vesica Piscis);

• \(s = 1\) — perpendicularity (\(90^\circ\) orthogonal poles);

• \(s = \frac{5 \mp \sqrt{5}}{8}\) — pentagonal / decagonal golden inclinations (\(36^\circ\) and \(72^\circ\)).

import sympy as sp
        import sympy.geometry as spg
        from geometor.model.invariants import line_spread

        l_base = spg.Line2D((0, 0), (1, 0))
        l_vesica = spg.Line2D((0, 0), (sp.Rational(1, 2), sp.sqrt(3) / 2))
        assert line_spread(l_base, l_vesica) == sp.Rational(3, 4)