W.03 Quadrance, Midpoints & Point-to-Line Separation

Replacing Irrational Euclidean Distance with Exact Quadratic Point Separation

The quadrance between A_1 = [x_1, y_1] and A_2 = [x_2, y_2] is the quadratic form Q(A_1, A_2) = (x_2 - x_1)^2 + (y_2 - y_1)^2; midpoint M divides the quadrance by 4, the locus Q(X, A_1) = Q(X, A_2) is the linear perpendicular bisector, and the quadrance from A_0 to a non-null line l = <a : b : c> is (a*x_0 + b*y_0 + c)^2 / (a^2 + b^2).

Interactive GEOMETOR construction for W.03 (Quadrance, Midpoints & Point-to-Line Separation). Hover any geometric label in the text or figure to highlight the corresponding element.

A_1 : Point A_2 : Point A_0 : Point ===== l = line(A_1, A_2) M = midpoint(A_1, A_2) b_12 = perp_bisector(A_1, A_2) F_0 = foot(A_0, l)

Mathematical Synthesis

Quadrance is the primary point-separation invariant of Rational Trigonometry (Wildberger2005). Because \(Q(A_1, A_2)\) avoids square roots, points in \(F^2\) always have separation in \(F\). Scaling a displacement by \(\lambda\) multiplies its quadrance by \(\lambda^2\), so the midpoint M of A1 and A2 satisfies \(Q(A_1, M) = Q(M, A_2) = Q(A_1, A_2)/4\).

Equating quadrances \(Q(X, A_1) = Q(X, A_2)\) cancels the quadratic terms \(x^2 + y^2\), yielding the linear perpendicular bisector b12 even when \(Q(A_1, A_2)\) is not a square in \(F\)! Similarly, the quadrance from an external point A0 to line l equals \(Q(A_0, F_0)\), and in any parallelogram the diagonal quadrances sum to twice the sum of adjacent side quadrances. Over fields where \(-1 = i^2\) (such as \(\mathbb{F}_{13}\) with \(5^2 \equiv -1\)), non-coincident points along a null line satisfy \(Q(A_1, A_2) = 0\).

Key Algebraic Identities

\[Q(A_1, A_2) = (x_2 - x_1)^2 + (y_2 - y_1)^2, \qquad Q(A_1, M) = Q(M, A_2) = \frac{Q(A_1, A_2)}{4}\]

\[Q(A_0, \ell) = \frac{(a x_0 + b y_0 + c)^2}{a^2 + b^2}, \qquad Q_{13} + Q_{24} = 2(Q_{12} + Q_{23})\]

Symbolic SymPy Verification

Invariant / Theorem Check	Exact SymPy Value	Verification Status
Segment quadrance Q(A_1, A_2)	40	No square root of 40 needed
Midpoint quadrance Q(A_1, M) = Q/4	10	Verified 4 * 10 == 40
Perpendicular bisector b_12	<-6 : -2 : 28>	Linear locus Q(X,A_1)=Q(X,A_2)
Point-to-line quadrance Q(A_0, l)	98/5	Matches Q(A_0, F_0) = 98/5
Null segment [0,0]->[1,5] over GF(13)	1^2 + 5^2 = 26 == 0 (mod 13)	Non-trivial isotropic line

Cross-Repository Canon

Executable Demonstration	W.03 Quadrance, Midpoints & Point-to-Line Separation — Code (Standalone model.py & w-03.svg)
Thematic Part	Part I: Preliminaries & Foundational Invariants
Euclid’s Elements	I.10, I.47, II.4 (EuclidElements)
GEOMETOR Elements	G.1 point, G.3 extremities of a segment
GEOMETOR Fundamentals	F.9 Circle Equations
Sister Codices	geometor/model (exact polynomial distance/quadrance algebra)

Constructive Dependency Graph

Depends on: W.01 Planar Points, Line Proportions & Incidence Determinants

Required for: W.04 Spread, Cross, Twist & Line Separation Ratios, W.06 Law I: Pythagoras’ Theorem & Right Triangle Invariants, W.07 Law II: Triple Quad Formula, Archimedes’ Function & Quadrea, W.12 Vertex Bisectors, Square Vertices & The Four Incenters, W.13 Stewart’s Theorem, Median Quadrances & The Centroid, W.15 Circumcenter, Orthocenter & Universal Affine Barycentrics, W.20 Metrical Conics: Circles, Ribbons, Parabolas, Quadrolas & Grammolas, W.21 Taylor Conics, Algebraic Tangents & Optical Reflection Laws, W.22 Projective Cross-Ratio, Harmonic Pencils & The Pappus Bridge, W.28 Rational Polar Coordinates, Classical Curves & The Ellipson