T_24 • Equilateral, Isogonic & Napoleon Triangles

Category: Derived Central Triangles

The outer and inner equilateral triangles erected on sides BC, CA, AB, whose apexes form the Isogonic Triangles (perspectors X₁₃, X₁₄) and whose centers form the equilateral Napoleon Triangles (perspectors X₁₇, X₁₈).

Primary Construction & Central Lines

T_24 • Equilateral, Isogonic & Napoleon Triangles Toggle Central Lines

Key Properties & Sequenced Operations

1. Definition (NAPOLEON’S THEOREM): Isogonic Triangles (Perspectors X_13, X_14) • Napoleon Triangles (Perspectors X_17, X_18) — Equilateral Constructions: Outer and inner equilateral apexes yield Fermat points X₁₃, X₁₄, while their centroids form equilateral triangles with perspectors X₁₇, X₁₈ on the circumcircle stage. (For any triangle ABC, the centers of outer (or inner) equilateral triangles on its sides form an equilateral triangle.)

2. Coordinate & Algebraic Law (TORRICELLI CIRCLES): Area(Outer Napoleon) − Area(Inner Napoleon) = Area(△ABC)  •  Shared Centroid X_2 — Area & Centroid Invariance: Both Napoleon triangles have centroid X₂, while X₁₃, X₁₄ and X₁₇, X₁₈ correspond to Kiepert angles ±60° and ±30°. (The circumcircles of the three outer equilateral triangles concur at the 1st Fermat point X₁₃.)

3. Central Line Incidences (COLLINEAR EQUILATERAL CENTERS): Kiepert Chords {2, 13, 17} • {4, 13, 18} • {3, 17, 18} — Napoleon Central Lines: Centroid axis {2, 13, 17}, orthocenter axis {4, 13, 18}, and circumcenter axis {3, 17, 18}. (X₁₇ and X₁₈ are isogonal conjugates of each other’s pedal/circumcircle partners on the Kiepert hyperbola.)

4. Collection 1 Network (COLLECTION 1 CAPSTONE): X_17 • X_18 • X_13 • X_14 • X_15 • X_16 — Equilateral Six-Center Family: Napoleon points X₁₇, X₁₈, Fermat points X₁₃, X₁₄, and Isodynamic points X₁₅, X₁₆. (Completes the equilateral geometry of Collection 1 before Clawson point X₁₉.)

Collection 1 Centers Involved

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