W.06 Law I: Pythagoras’ Theorem & Right Triangle Invariants

Equivalence of Perpendicularity (s_3 = 1) and Additive Quadrances (Q_1 + Q_2 = Q_3)

For any three non-collinear points A_1, A_2, A_3 over a field F (char F != 2) with non-null lines A_1 A_3 and A_2 A_3, the lines A_1 A_3 and A_2 A_3 are perpendicular (s_3 = 1) if and only if Q(A_2, A_3) + Q(A_1, A_3) = Q(A_1, A_2), in which case the hypotenuse midpoint M_3 is equidistant in quadrance (Q_3 / 4) from all three vertices.

Interactive GEOMETOR construction for W.06 (Law I: Pythagoras’ Theorem & Right Triangle Invariants). Hover any geometric label in the text or figure to highlight the corresponding element.

A_1 : Point A_2 : Point A_3 : Point ===== t = polygon(A_1, A_2, A_3) M_3 = midpoint(A_1, A_2) c = circle(M_3, A_3) F_3 = foot(A_3, line(A_1, A_2))

Mathematical Synthesis

In Euclid’s Elements (I.47 and I.48), Pythagoras’ theorem equates geometric areas of squares erected on the sides of a right triangle. In Rational Trigonometry (Wildberger2005), squares on the sides are the native metric: \(Q_1 + Q_2 = Q_3\) is an exact linear equation in the three side quadrances whenever the lines at A3 have spread \(s_3 = 1\).

Two immediate corollaries hold over any field \(F\): the Right Quadrea Theorem \(\mathcal{A} = 4 Q_1 Q_2\), and the Right Midpoint Theorem showing that the midpoint M3 of hypotenuse \(A_1 A_2\) is the circumcenter of t with \(Q(M_3, A_1) = Q(M_3, A_2) = Q(M_3, A_3) = Q_3 / 4\), placing A3 on circle c and splitting the hypotenuse at altitude foot F3 in ratio \(Q_2 : Q_1\).

Key Algebraic Identities

\[\ell_1 \perp \ell_2 \iff s_3 = 1 \iff Q_1 + Q_2 = Q_3\]

\[Q(M_3, A_1) = Q(M_3, A_2) = Q(M_3, A_3) = \frac{Q_3}{4}, \qquad \mathcal{A}(Q_1, Q_2, Q_1 + Q_2) = 4 Q_1 Q_2\]

Symbolic SymPy Verification

Invariant / Theorem Check	Exact SymPy Value	Verification Status
Side quadrances (Q_1, Q_2, Q_3)	(16, 9, 25)	16 + 9 = 25
Right spread s_3 at A_3	1	l_1 perp l_2 <=> s_3 = 1
Right midpoint quadrance Q(M_3, A_i)	25/4	Equals Q_3 / 4 = 25/4
Right quadrea A = 4*Q_1*Q_2	576	Matches 4 * 16 * 9 = 576
Altitude foot F_3 split Q(A_1,F_3):Q(F_3,A_2)	81/25 : 256/25	Proportional to Q_2^2 : Q_1^2 = 81 : 256

Cross-Repository Canon

Executable Demonstration	W.06 Law I: Pythagoras’ Theorem & Right Triangle Invariants — Code (Standalone model.py & w-06.svg)
Thematic Part	Part II: The Five Fundamental Laws
Euclid’s Elements	I.47, I.48, III.31, VI.8 (EuclidElements)
GEOMETOR Elements	G.3 extremities of a segment, G.4 segment
GEOMETOR Fundamentals	F.10 Circles and Geometric Mean, F.42 Right Triangles
ETC Reference	Egyptian right triangle (3, 4, 5) -> Q = (9, 16, 25) in geometor/etc

Constructive Dependency Graph

Depends on: W.03 Quadrance, Midpoints & Point-to-Line Separation, W.04 Spread, Cross, Twist & Line Separation Ratios

Required for: W.07 Law II: Triple Quad Formula, Archimedes’ Function & Quadrea, W.08 Law III: The Spread Law & Extended Circumquadrance Law, W.09 Law IV: The Cross Law & Quadrea-Spread Identity, W.11 Isosceles, Equilateral & Kepler’s Golden Right Triangle, W.16 The Euler Line & Nine-Point Circle Over Arbitrary Fields, W.17 Circle Geometry: Subtended Spreads, Secant Products & Rational Points, W.23 Chromogeometry & Universal Symmetric Bilinear Forms, W.27 3D Planes, De Gua’s Quadrea Law & Platonic Dihedral Spreads