Principles

The GEOMETOR Wildberger Codex gathers 28 foundational principles (\(\text{W.01}\) through \(\text{W.28}\)) of Rational Trigonometry and Universal Geometry (Wildberger2005, Wildberger2008), demonstrating each theorem through 100% exact SymPy symbolic proofs, runnable geometor.model constructions, and interactive SVG diagrams.

• Part I (W.01 – W.05): Planar proportions, reflections, quadrance, spread, and spread polynomials \(S_n(s)\) (including Pentagonal Golden Spreads).

• Part II (W.06 – W.11): The Five Fundamental Laws (Pythagoras, Triple Quad, Spread Law, Cross Law, Triple Spread) and Kepler’s Golden Right Triangle.

• Part III (W.12 – W.23): Triangle centers, Euler line & Nine-Point Circle over \(\mathbb{Q}\) and \(\mathbb{F}_p\), circles, cyclic quadrilaterals, golden pentagons, metrical conics, Taylor tangents, projective cross-ratio, and Chromogeometry (Blue, Red, Green metrics).

• Part IV (W.24 – W.28): Polynomial triangle solvers, Snellius–Pothenot resection, Hansen’s surveying benchmark, algebraic physics, exact rational dihedral spreads of all five Platonic solids, rational polar curves, and the Ellipson.

W.01 Planar Points, Line Proportions & Incidence Determinants

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.

W.02 Lineations, Central Rotations & Reflections in Lines

For any non-null line l = <a : b : c> over a field F (char F != 2), the reflection sigma_l maps every point A = [x, y] and every line m = <a_2 : b_2 : c_2> to exact rational images whose denominators depend only on the quadratic norm a^2 + b^2, without radicals or angles.

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

W.04 Spread, Cross, Twist & Line Separation Ratios

For two non-null lines l_1 = <a_1 : b_1 : c_1> and l_2 = <a_2 : b_2 : c_2>, the spread s(l_1, l_2) = (a_1*b_2 - a_2*b_1)^2 / ((a_1^2 + b_1^2)*(a_2^2 + b_2^2)) depends only on the unoriented lines, vanishes iff l_1 || l_2, equals 1 iff l_1 perp l_2, and equals Q(B, C) / Q(A, B) for any altitude BC dropped from B on l_1 to C on l_2.

W.05 Spread Polynomials S_n(s) & Pentagonal Golden Spreads

The integer-coefficient spread polynomials S_0(s) = 0, S_1(s) = s, S_n(s) = 2(1 - 2s)S_{n-1}(s) - S_{n-2}(s) + 2s satisfy the semigroup composition law S_n(S_m(s)) = S_{nm}(s) over every commutative ring, and the fivefold polynomial S_5(s) = s(16s^2 - 20s + 5)^2 vanishes precisely at the Pentagonal Golden Spreads s = (5 -/+ sqrt(5))/8 = (3 - phi)/4 and (2 + phi)/4.

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

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.

W.07 Law II: Triple Quad Formula, Archimedes’ Function & Quadrea

Three planar points A_1, A_2, A_3 with pairwise quadrances Q_1, Q_2, Q_3 are collinear if and only if Archimedes’ symmetric quadratic form A(Q_1, Q_2, Q_3) = (Q_1 + Q_2 + Q_3)^2 - 2(Q_1^2 + Q_2^2 + Q_3^2) vanishes; for any triangle, A(Q_1, Q_2, Q_3) equals 4 times the squared coordinate determinant (16 * Delta^2), defining the triangle’s quadrea.

W.08 Law III: The Spread Law & Extended Circumquadrance Law

In any non-null triangle A_1 A_2 A_3 with side quadrances Q_1, Q_2, Q_3, vertex spreads s_1, s_2, s_3, quadrea A, and circumquadrance K, the ratios of spread to opposite quadrance are equal: s_1 / Q_1 = s_2 / Q_2 = s_3 / Q_3 = A / (4 * Q_1 * Q_2 * Q_3) = 1 / (4 * K).

W.09 Law IV: The Cross Law & Quadrea-Spread Identity

In any triangle A_1 A_2 A_3 with non-null sides Q_1, Q_2 meeting with spread s_3 and cross c_3 = 1 - s_3 at vertex A_3, the quadrances satisfy (Q_1 + Q_2 - Q_3)^2 = 4 * Q_1 * Q_2 * (1 - s_3) = 4 * Q_1 * Q_2 * c_3, equivalently A(Q_1, Q_2, Q_3) = 4 * Q_1 * Q_2 * s_3.

W.10 Law V: Triple Spread, Triple Cross & Quadruple Spread Formulas

For any three coplanar non-null lines l_1, l_2, l_3 with pairwise spreads s_1, s_2, s_3 and crosses c_i = 1 - s_i, the spreads satisfy the cubic Triple Spread Formula (s_1 + s_2 + s_3)^2 = 2(s_1^2 + s_2^2 + s_3^2) + 4*s_1*s_2*s_3 and the crosses satisfy (c_1 + c_2 + c_3 - 1)^2 = 4*c_1*c_2*c_3; for four lines, the Quadruple Spread polynomial R(s_12, s_23, s_34, s_14) = 0 is identically invariant under s_ij <-> 1 - s_ij.

W.11 Isosceles, Equilateral & Kepler’s Golden Right Triangle

In a non-null triangle, Q_1 = Q_2 iff s_1 = s_2 (with apex spread s_3 = (Q_3/Q_1)(1 - Q_3/(4*Q_1))), and Q_1 = Q_2 = Q_3 iff s_1 = s_2 = s_3 = 3/4; in Kepler’s Golden Right Triangle with side lengths (1, sqrt(phi), phi), the nested radical vanishes in quadrance to give Q = (1, phi, phi^2) and exact golden spreads (s_1, s_2, s_3) = (2 - phi, phi - 1, 1) = (1/phi^2, 1/phi, 1).

W.12 Vertex Bisectors, Square Vertices & The Four Incenters

The locus of points X having equal quadrance Q(X, l_1) = Q(X, l_2) to two intersecting non-null lines l_1, l_2 splits into two orthogonal vertex bisectors over F if and only if (a_1^2 + b_1^2)(a_2^2 + b_2^2) is a square in F; when two vertices of a triangle have bisectors in F, so does the third, producing four symmetric incenters I_0, I_1, I_2, I_3.

W.13 Stewart’s Theorem, Median Quadrances & The Centroid

For any affine point D = (1 - lambda)*A_1 + lambda*A_2 on side A_1 A_2 of triangle A_1 A_2 A_3, the cevian quadrance satisfies Q(A_3, D) = (1 - lambda)*Q_2 + lambda*Q_1 - lambda*(1 - lambda)*Q_3; at lambda = 1/2, the median quadrance satisfies 4*Q(A_3, M_3) = 2*Q_1 + 2*Q_2 - Q_3, and the three medians concur at G = (A_1 + A_2 + A_3)/3.

W.14 Menelaus, Ceva & The Alternate Spreads Theorem

If either a transversal line l cuts the sidelines A_2 A_3, A_3 A_1, A_1 A_2 of a non-null triangle at B_1, B_2, B_3 (Menelaus) or three cevians through a point A_0 meet the opposite sidelines at B_1, B_2, B_3 (Ceva), then the six segment quadrances satisfy Q(A_2, B_1) * Q(A_3, B_2) * Q(A_1, B_3) = Q(B_1, A_3) * Q(B_2, A_1) * Q(B_3, A_2).

W.15 Circumcenter, Orthocenter & Universal Affine Barycentrics

Every non-null triangle A_1 A_2 A_3 over a field F (char F != 2) has a unique circumcenter C at the intersection of its side perpendicular bisectors and a unique orthocenter O at the intersection of its altitudes, with affine barycentric weights beta_i = Q_i*(Q_j + Q_k - Q_i)/(2*A) for C and gamma_i = (Q_i + Q_j - Q_k)*(Q_k + Q_i - Q_j)/(2*A) for O.

W.16 The Euler Line & Nine-Point Circle Over Arbitrary Fields

In any non-null triangle A_1 A_2 A_3 over a field F (char F != 2), the nine-point center N = (C + O)/2 satisfies 2*N - C = O (and 3*G = 2*C + O when char F != 3), placing C, G, N, O on the Euler line with quadrance proportions Q(C, G) : Q(C, N) : Q(C, O) = 4 : 9 : 36; the three side midpoints M_i, three altitude feet F_i, and three Euler midpoints E_i = (A_i + O)/2 all have exact quadrance K / 4 from N.

W.17 Circle Geometry: Subtended Spreads, Secant Products & Rational Points

On a circle Q(X, O_c) = K over F, a diameter subtends spread 1 at every other circle point (Thales), any chord P_1 P_2 subtends constant spread s_3 at the circumference with Q(P_1, P_2) = 4*K*s_3 and central spread s(O_c P_1, O_c P_2) = S_2(s_3) = 4*s_3*(1 - s_3), and any external or internal point A_0 has invariant secant quadrance product Q(A_0, P_1)*Q(A_0, P_2) = (Q(A_0, O_c) - K)^2.

W.18 Cyclic Quadrilaterals, Ptolemy & Euler’s Four-Point Relation

Four non-collinear points A_1, A_2, A_3, A_4 on a circle c satisfy equal opposite spreads s(A_1 A_2, A_1 A_4) = s(A_3 A_2, A_3 A_4), Ptolemy’s Triple-Quad Law A(Q_12*Q_34, Q_23*Q_14, Q_13*Q_24) = 0 among product quadrances, and the cyclic quadrea / circumquadrance formulas, while four collinear points satisfy Q(Q_12, Q_23, Q_34, Q_14) = 0.

W.19 Regular Stars, Golden Pentagons & Finite-Field Polygons

A regular n-star of lines has consecutive spread s satisfying S_n(s) = 0; for n = 5, a regular pentagon P_0..P_4 has side-to-diagonal spread alpha = (5 - sqrt(5))/8 = (3 - phi)/4, exterior/star spread beta = (5 + sqrt(5))/8 = (2 + phi)/4, and exact golden quadrance ratio Q_diag / Q_side = beta / alpha = phi^2 = phi + 1, existing rationally in F_p^2 whenever 5 and -1/5-dependent discriminants split (e.g. regular 5-stars in F_19).

W.20 Metrical Conics: Circles, Ribbons, Parabolas, Quadrolas & Grammolas

A conic is the zero locus of a quadratic polynomial proportion C = <a:b:c:d:e:f>; metrically, Q(X, l) = R defines a ribbon, Q(X, F_0) = Q(X, l_dir) defines a parabola, the squared-sum/difference bifocal locus A(Q(X, F_1), Q(X, F_2), S) = 0 defines a quadrola (unifying ellipses and hyperbolas into one irreducible algebraic curve), and Q(X, l_1) +/- Q(X, l_2) = R defines a grammola.

W.21 Taylor Conics, Algebraic Tangents & Optical Reflection Laws

Translating any conic f(x, y) = 0 to a point A = [x_0, y_0] on the conic isolates a unique linear term l_1(x - x_0, y - y_0) = 0 defining its algebraic tangent t_tan without limits; for a parabola, t_tan is the perpendicular bisector of focus F_0 and directrix foot D_0 (giving equal spreads with F_0 A and the axis), and for a quadrola, t_tan makes equal spreads with both focal lines F_1 P_q and F_2 P_q.

W.22 Projective Cross-Ratio, Harmonic Pencils & The Pappus Bridge

If four concurrent lines r_1, r_2, r_3, r_4 through O_p meet a transversal l_a at collinear points A_1, A_2, A_3, A_4 and another transversal l_b at B_1, B_2, B_3, B_4, then the squared cross-ratio in quadrances (Q(A_1, A_3)*Q(A_2, A_4)) / (Q(A_1, A_4)*Q(A_2, A_3)) equals the pencil spread cross-ratio (s_13 * s_24) / (s_14 * s_23) and is invariant across all transversals.

W.23 Chromogeometry & Universal Symmetric Bilinear Forms

Replacing the Euclidean quadratic form with any non-degenerate symmetric bilinear form M on F^2 preserves all Five Laws of Rational Trigonometry; in Chromogeometry—pairing Blue (dx^2 + dy^2), Red (dx^2 - dy^2), and Green (2*dx*dy) metrics on the same triangle—the three Euler lines concur at the centroid G, C_B C_R O_B C_G is a parallelogram, and N_B = (C_R + C_G)/2.

W.24 Exact Triangle Solvers & Surveying Triangulation

Every classical triangle specification (QQQ, QQS, QSS, SQQ, and SSS up to scaling) reduces to linear or quadratic equations in quadrances and spreads with exact discriminant branching; applied to surveying, inaccessible heights and Regiomontanus’ maximum-spread viewing problem resolve by pure rational algebra without trigonometric tables.

W.25 Snellius–Pothenot Resection & Hansen’s Surveying Problem

In Snellius-Pothenot resection, an observer P measuring spreads to three known beacons A, B, C lies on the algebraic intersection of two subtended-spread conics; in Hansen’s problem, knowing one baseline quadrance Q(A, B) and four sighting spreads at unobserved stations C, D determines every side and diagonal quadrance exactly via Triple Spread Systems and Alternate Spreads.

W.26 Algebraic Physics: Projectile Parabolas, Snell’s Law & Relativity

A Galilean projectile launched from O with initial speed-squared v^2 and elevation spread s traces a parabola whose apex height h = v^2*s/(2*g), horizontal range quadrance Q_range = 16*(v^4/(4*g^2))*s*(1 - s) = (v^4/g^2)*S_2(s), and focus-directrix geometry depend rationally on s; Snell’s refraction law becomes n_1^2 * s_1 = n_2^2 * s_2 in spreads.

W.27 3D Planes, De Gua’s Quadrea Law & Platonic Dihedral Spreads

In 3D space over F, quadrance is Q(A_1, A_2) = dx^2 + dy^2 + dz^2, a right-corner tetrahedron satisfies De Gua’s Quadrea Theorem A_base = A_xy + A_yz + A_zx, and the dihedral (face) spreads of ALL FIVE regular Platonic solids—Tetrahedron (8/9), Cube (1), Octahedron (8/9), Icosahedron (4/9), and Dodecahedron (4/5)—are exact rational numbers in Q!

W.28 Rational Polar Coordinates, Classical Curves & The Ellipson

Replacing polar coordinates (r, theta) with rational polar coordinates [s, Q] where s = x^2 / (x^2 + y^2) (spread to the y-axis) and Q = x^2 + y^2 converts classical polar curves into exact polynomial relations—Bernoulli’s lemniscate s = (Q + 2)/4 (or Q = a*(2*s - 1)), Descartes’ folium (2*s - 1)^2*(s^2 - s + 1)^2*Q^2 - 18*s*(1 - s)*(1 - 3*s + 3*s^2)*Q + 81*s^2*(1 - s)^2 = 0, and the deltoid/ellipson S_3(s) law—while (s_1 + s_2 + s_3)^2 = 2(s_1^2 + s_2^2 + s_3^2) + 4*s_1*s_2*s_3 cuts out the tetrahedral Ellipson.

Master Index of Principles (W.01 – W.28)

Code	Principle	Category	Thematic Division
W.01	Planar Points, Line Proportions & Incidence Determinants	Affine & Projective Incidence	Part I: Preliminaries & Foundational Invariants
W.02	Lineations, Central Rotations & Reflections in Lines	Isometries & Lineations	Part I: Preliminaries & Foundational Invariants
W.03	Quadrance, Midpoints & Point-to-Line Separation	Foundational Quadratic Invariants	Part I: Preliminaries & Foundational Invariants
W.04	Spread, Cross, Twist & Line Separation Ratios	Foundational Quadratic Invariants	Part I: Preliminaries & Foundational Invariants
W.05	Spread Polynomials S_n(s) & Pentagonal Golden Spreads	Polynomial Families & Golden Proportion	Part I: Preliminaries & Foundational Invariants
W.06	Law I: Pythagoras’ Theorem & Right Triangle Invariants	The Five Fundamental Laws	Part II: The Five Fundamental Laws
W.07	Law II: Triple Quad Formula, Archimedes’ Function & Quadrea	The Five Fundamental Laws	Part II: The Five Fundamental Laws
W.08	Law III: The Spread Law & Extended Circumquadrance Law	The Five Fundamental Laws	Part II: The Five Fundamental Laws
W.09	Law IV: The Cross Law & Quadrea-Spread Identity	The Five Fundamental Laws	Part II: The Five Fundamental Laws
W.10	Law V: Triple Spread, Triple Cross & Quadruple Spread Formulas	The Five Fundamental Laws	Part II: The Five Fundamental Laws
W.11	Isosceles, Equilateral & Kepler’s Golden Right Triangle	Special & Golden Triangles	Part II: The Five Fundamental Laws
W.12	Vertex Bisectors, Square Vertices & The Four Incenters	Triangle Centers & Number Theory	Part III: Universal Triangle, Conic & Chromogeometry
W.13	Stewart’s Theorem, Median Quadrances & The Centroid	Proportions, Cevians & Medians	Part III: Universal Triangle, Conic & Chromogeometry
W.14	Menelaus, Ceva & The Alternate Spreads Theorem	Projective & Cevian Proportions	Part III: Universal Triangle, Conic & Chromogeometry
W.15	Circumcenter, Orthocenter & Universal Affine Barycentrics	Triangle Centers & Classical Lines	Part III: Universal Triangle, Conic & Chromogeometry
W.16	The Euler Line & Nine-Point Circle Over Arbitrary Fields	Triangle Centers & Classical Circles	Part III: Universal Triangle, Conic & Chromogeometry
W.17	Circle Geometry: Subtended Spreads, Secant Products & Rational Points	Circle Geometry & Rational Parametrization	Part III: Universal Triangle, Conic & Chromogeometry
W.18	Cyclic Quadrilaterals, Ptolemy & Euler’s Four-Point Relation	Cyclic Polygons & Four-Point Invariants	Part III: Universal Triangle, Conic & Chromogeometry
W.19	Regular Stars, Golden Pentagons & Finite-Field Polygons	Regular Polygons, Golden Stars & Finite Fields	Part III: Universal Triangle, Conic & Chromogeometry
W.20	Metrical Conics: Circles, Ribbons, Parabolas, Quadrolas & Grammolas	Metrical Conics & Quadratic Loci	Part III: Universal Triangle, Conic & Chromogeometry
W.21	Taylor Conics, Algebraic Tangents & Optical Reflection Laws	Algebraic Tangents & Optical Reflection	Part III: Universal Triangle, Conic & Chromogeometry
W.22	Projective Cross-Ratio, Harmonic Pencils & The Pappus Bridge	Projective Geometry & Harmonic Ranges	Part III: Universal Triangle, Conic & Chromogeometry
W.23	Chromogeometry & Universal Symmetric Bilinear Forms	Universal Bilinear Forms & Chromogeometry	Part III: Universal Triangle, Conic & Chromogeometry
W.24	Exact Triangle Solvers & Surveying Triangulation	Practical Triangle Solvers & Surveying	Part IV: Practical Solvers, 3D Platonic Geometry & Polar Curves
W.25	Snellius–Pothenot Resection & Hansen’s Surveying Problem	Geodetic Resection & Two-Station Surveying	Part IV: Practical Solvers, 3D Platonic Geometry & Polar Curves
W.26	Algebraic Physics: Projectile Parabolas, Snell’s Law & Relativity	Algebraic Mechanics, Optics & Relativity	Part IV: Practical Solvers, 3D Platonic Geometry & Polar Curves
W.27	3D Planes, De Gua’s Quadrea Law & Platonic Dihedral Spreads	3D Solid Geometry & The Five Platonic Solids	Part IV: Practical Solvers, 3D Platonic Geometry & Polar Curves
W.28	Rational Polar Coordinates, Classical Curves & The Ellipson	Rational Polar Coordinates & Algebraic Surfaces	Part IV: Practical Solvers, 3D Platonic Geometry & Polar Curves