Glossary of Geometric & Algebraic Terms¶
Definitions of classical Euclidean, projective, and Rational Trigonometry concepts used throughout the catalog.
- Golden Ratio (φ) (Sectio Aurea / Divina Proportione)
The positive root φ = (1 + √5)/2 ≈ 1.6180339887 of x² − x − 1 = 0, satisfying φ² = φ + 1 and 1/φ = φ − 1. Appears in: GC-03.01, GC-03.02, GC-03.03, GC-03.04, GC-03.05, GC-03.06, GC-03.07, GC-03.08, GC-04.01, GC-04.02, GC-04.03, GC-04.04 (+19 more)
- Golden Section (Collinear Division in φ)
A collinear point triple (A, G, B) whose consecutive segments stand in the golden ratio |AG|/|GB| = φ (so |AB|/|AG| = φ as well). Appears in: GC-01.01, GC-01.02, GC-01.03, GC-01.04, GC-01.06, GC-01.07, GC-02.01, GC-02.02, GC-02.03, GC-02.04, GC-02.05, GC-02.06 (+32 more)
- Extreme and Mean Ratio (ἄκρος καὶ μέσος λόγος (Euclid VI.Def.3))
Euclid’s classical definition: a straight line is cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less. Appears in: GC-01.01, GC-05.01
- Golden Chain (Connected Collinear φ-Progression)
A sequence of four or more collinear points in which overlapping triples form consecutive golden sections (such as lengths 1/φ, 1, 1/φ or 1, φ, φ²). Appears in: GC-01.02, GC-01.03, GC-01.05, GC-02.01, GC-02.07, GC-03.01, GC-03.07, GC-03.08, GC-05.01, GC-05.03, GC-05.05, GC-06.03 (+5 more)
- Quadrance (Q(A, B) = (x₂ − x₁)² + (y₂ − y₁)²)
Norman Wildberger’s Rational Trigonometry invariant measuring squared separation without square roots; two segments stand in ratio φ iff Q_long / Q_short = φ + 1 = φ². Appears in: GC-01.01, GC-01.02, GC-01.07, GC-02.02, GC-02.03, GC-02.04, GC-02.06, GC-03.03, GC-03.04, GC-03.08, GC-04.02, GC-04.03 (+2 more)
- Géométrographie (Lemoine Construction Complexity)
Émile Lemoine’s 1902 operational metric counting elementary straightedge (R₁, R₂) and compass (C₁, C₂, C₃) operations; drawing a line through two points or a circle centered at one point through another each contributes simplicity 3. Appears in: GC-01.03, GC-01.04, GC-01.07
- Vesica Piscis (Twin-Circle Lens (Euclid I.1))
The symmetric lens formed by two congruent circles each passing through the other’s center, generating √3 altitudes and bisecting their common radius. Appears in: GC-01.03, GC-01.04, GC-01.05, GC-01.06, GC-06.03
- Mascheroni Construction (Compass-Only Construction)
A geometric construction using circles alone (0 straightedge lines), proved universal for all Euclidean constructible points by Georg Mohr (1672) and Lorenzo Mascheroni (1797). Appears in: GC-01.05
- Rusty Compass (Fixed-Aperture Compass)
A compass locked to a single invariant radius throughout a construction, studied by Abu’l-Wafa, Leonardo da Vinci, Poncelet, and Steiner. Appears in: GC-01.06
- Power of a Point (Euclid III.35–37 / Radical Invariant)
For a point P and circle O(r), the invariant quantity |PO|² − r², equal to the signed product |PA|·|PB| along any secant PAB (and to |PT|² along a tangent PT). Appears in: GC-02.05, GC-04.01, GC-06.02, GC-06.06
- Intersecting Chords Theorem (Euclid Elements III.35)
If two chords AB and CD of a circle intersect internally at M, then |AM|·|MB| = |CM|·|MD|. Appears in: GC-02.01, GC-03.01, GC-06.04
- Butterfly Configuration (Crossed Cyclic Chords across a Midpoint Chord)
Cyclic quadrilateral diagonals crossing a reference chord symmetrically, inducing projective cross-ratio and golden-section divisions. Appears in: GC-02.07
- Kepler Triangle (1 : √φ : φ Right Triangle)
The unique right triangle whose side lengths (and side quadrances 1 : φ : φ²) form a geometric progression. Appears in: GC-04.02, GC-06.05
- Mixtilinear Incircle (Two-Side & Circumcircle Tangent Circle)
A circle internally tangent to two sides of a triangle and internally tangent to its circumcircle. Appears in: GC-07.05
- Golden Rhombus (Diagonal Ratio |d₁|/|d₂| = φ)
A rhombus whose long and short diagonals stand in the golden ratio φ; thirty congruent golden rhombi bound Kepler’s rhombic triacontahedron. Appears in: GC-07.02