T_9 • The Feuerbach Line

Category: Named Central Lines

The line of centers passing through the Incenter X₁, Nine-Point Center X₅, Feuerbach Point X₁₁, and its harmonic conjugate X₁₂ at the tangency of the incircle and nine-point circle.

Primary Construction & Central Lines

T_9 • The Feuerbach Line Toggle Central Lines

Key Properties & Sequenced Operations

1. Definition (FEUERBACH’S THEOREM (1822)): Feuerbach Line L(1, 5, 11, 12) • |X_1 X_5| = |R/2 − r| — The Feuerbach Line: Joins Incenter X₁ and Nine-Point Center X₅ through tangency point X₁₁ and X₁₂ of the incircle and nine-point circle. (Internal tangency requires the distance between centers |X₁ X₅| to equal the difference of radii |R/2 − r|.)

2. Coordinate & Algebraic Law (HARMONIC RANGE (X₁, X₅; X₁₁, X₁₂) = −1): Harmonic Similicenters: X_11 (Internal) & X_12 (External) — Tangency & Similitude: X₁₁ is the internal tangency point and X₁₂ is the harmonic conjugate dividing X₁ X₅ in ratio r : R/2. (X₁₁ and X₁₂ are the internal and external centers of similitude of the incircle and nine-point circle.)

3. Central Line Incidences (LINE OF CENTERS): Collinearity {1, 5, 11, 12} — Feuerbach Axis: The central line {1, 5, 11, 12} connects two tangent circles and four Collection 1 centers. (Perpendicular to the common tangent of the incircle and nine-point circle at X₁₁.)

4. Collection 1 Network (COLLECTION 1 HARMONIC PAIR): X_1 • X_5 • X_11 • X_12 — Feuerbach Quartet: X₁, X₅, X₁₁, and X₁₂ form a harmonic quartet on the Feuerbach line. (X₁₁ and X₁₂ are also perspectors of the orthic and intouch configurations.)

Collection 1 Centers Involved

X₁, X₅, X₁₁, X₁₂