W.01 Planar Points, Line Proportions & Incidence Determinants

Field-Independent Coordinate Incidence, Parallels, and Altitudes

Over any field F of characteristic not equal to 2, a planar point is an ordered pair A = [x, y] and a planar line is a homogeneous 3-proportion l = <a : b : c> with (a, b) != (0, 0) representing a*x + b*y + c = 0; three points are collinear and three lines are concurrent if and only if their respective 3x3 coordinate determinants vanish.

Interactive GEOMETOR construction for W.01 (Planar Points, Line Proportions & Incidence Determinants). Hover any geometric label in the text or figure to highlight the corresponding element.

A_1 : Point A_2 : Point A_3 : Point ===== l_3 = line(A_1, A_2) p_3 = parallel(l_3, A_3) n_3 = altitude(l_3, A_3) F_3 = meet(l_3, n_3)

Mathematical Synthesis

In Norman J. Wildberger’s algebraic foundation (Wildberger2005), classical continuum postulates are replaced by arithmetic over an arbitrary field \(F\) (\(\operatorname{char} F \neq 2\)). Two distinct points A1 \(= [x_1, y_1]\) and A2 \(= [x_2, y_2]\) determine a unique line proportion l3 \(= \langle y_1 - y_2 : x_2 - x_1 : x_1 y_2 - x_2 y_1 \rangle\), while two non-parallel lines meet at a unique rational point given by Cramer’s determinant ratios.

Parallelism (\(a_1 b_2 - a_2 b_1 = 0\)) and Euclidean perpendicularity (\(a_1 a_2 + b_1 b_2 = 0\)) are purely polynomial conditions. Given an external point A3 and a non-null line l3, both the parallel line p3 and the altitude n3 meeting l3 at its foot F3 are obtained by exact rational expressions without square roots or compass arcs—bridging René Descartes’ coordinate algebra directly to projective incidence in geometor/pappus.

Key Algebraic Identities

\[\ell(A_1, A_2) = \langle y_1 - y_2 : x_2 - x_1 : x_1 y_2 - x_2 y_1 \rangle, \qquad \ell_1 \cap \ell_2 = \left[\frac{b_1 c_2 - b_2 c_1}{a_1 b_2 - a_2 b_1},\; \frac{a_2 c_1 - a_1 c_2}{a_1 b_2 - a_2 b_1}\right]\]

\[\begin{split}F_3 = \left[\frac{b^2 x_3 - a b y_3 - a c}{a^2 + b^2},\; \frac{-a b x_3 + a^2 y_3 - b c}{a^2 + b^2}\right], \qquad \det\begin{pmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{pmatrix} = 0\end{split}\]

Symbolic SymPy Verification

Invariant / Theorem Check	Exact SymPy Value	Verification Status
Line proportion l_3 = line(A_1, A_2)	<-3 : 6 : -6>	Exact integer 3-proportion
Parallel proportion p_3 through A_3	<-3 : 6 : -30>	a_1*b_2 - a_2*b_1 = 0
Altitude foot F_3 = meet(l_3, n_3)	[18/5, 14/5]	Zero radicals
Collinearity det(A_1, A_2, F_3)	0	Verified == 0
Concurrency det at F_3	0	Verified == 0

Cross-Repository Canon

Executable Demonstration	W.01 Planar Points, Line Proportions & Incidence Determinants — Code (Standalone model.py & w-01.svg)
Thematic Part	Part I: Preliminaries & Foundational Invariants
Euclid’s Elements	I.11, I.12, I.31 (EuclidElements)
GEOMETOR Elements	G.1 point, G.3 extremities of a segment, G.4 segment
GEOMETOR Fundamentals	F.14 Conway Notation (SeeligNotebooks)
Sister Codices	geometor/pappus (projective collinearity & concurrency determinants)

Constructive Dependency Graph

Required for: W.02 Lineations, Central Rotations & Reflections in Lines, W.03 Quadrance, Midpoints & Point-to-Line Separation, W.04 Spread, Cross, Twist & Line Separation Ratios, W.07 Law II: Triple Quad Formula, Archimedes’ Function & Quadrea, W.13 Stewart’s Theorem, Median Quadrances & The Centroid, W.14 Menelaus, Ceva & The Alternate Spreads Theorem, W.15 Circumcenter, Orthocenter & Universal Affine Barycentrics, W.22 Projective Cross-Ratio, Harmonic Pencils & The Pappus Bridge