T_13 • The Isogonic & Napoleon Central Lines

Category: Named Central Lines

The central axes generated by outer and inner equilateral triangles on the sides of △ABC, linking Fermat points X₁₃, X₁₄, Isodynamic points X₁₅, X₁₆, and Napoleon points X₁₇, X₁₈.

Primary Construction & Central Lines

T_13 • The Isogonic & Napoleon Central Lines Toggle Central Lines

Key Properties & Sequenced Operations

1. Definition (TORRICELLI, FERMAT, APOLLONIUS & NAPOLEON): Equilateral Triad: Fermat (X_13, X_14) • Isodynamic (X_15, X_16) • Napoleon (X_17, X_18) — Equilateral Central Lines: Fermat points X₁₃, X₁₄, Isodynamic points X₁₅, X₁₆, and Napoleon points X₁₇, X₁₈ on the circumcircle stage. (All six centers arise from the π/3 and −π/3 Kiepert equilateral constructions on sides BC, CA, AB.)

2. Coordinate & Algebraic Law (60° & 30° KIEPERT PARAMETERS): Kiepert Hyperbola {2, 4, 13, 14, 17, 18} ↔ Brocard Axis {3, 6, 15, 16} — Kiepert–Brocard Duality: X₁₃, X₁₄, X₁₇, X₁₈ lie on the Kiepert hyperbola with X₂, X₄, while their isogonal partners X₁₅, X₁₆ lie on the Brocard axis. (Kiepert hyperbola K(θ) gives X₁₃, X₁₄ at θ = ±60° and X₁₇, X₁₈ at θ = ±30°.)

3. Central Line Incidences (CROSS-ALIGNMENT OF TRIADS): Collinearities {3, 6, 15, 16} • {2, 13, 17} • {4, 13, 18} — Equilateral Axes: Brocard axis {3, 6, 15, 16}, centroid–Fermat–Napoleon line {2, 13, 17}, and orthocenter–Fermat–Napoleon line {4, 13, 18}. (Line {6, 13, 16} and {6, 14, 15} also cross at the Symmedian point X₆.)

4. Collection 1 Network (COMPLETE EQUILATERAL SUBSYSTEM): X_13 ↔ X_15 • X_14 ↔ X_16 • X_17 ↔ X_18 — Isogonal Pairs: X₁₃ and X₁₅, X₁₄ and X₁₆, and X₁₇ and X₁₈. (Together X₁₃–X₁₈ form a closed subsystem under isogonal conjugation and circumcircle inversion.)

Collection 1 Centers Involved

X₂, X₃, X₄, X₆, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈