T_11 • The Soddy Line

Category: Named Central Lines

The central line passing through the Incenter X₁, Gergonne Point X₇, and de Longchamps Point X₂₀, carrying the inner and outer Soddy centers X₁₇₅ and X₁₇₆.

Primary Construction & Central Lines

T_11 • The Soddy Line Toggle Central Lines

Key Properties & Sequenced Operations

1. Definition (FREDERICK SODDY (1936) • DESCARTES CIRCLE THEOREM): Soddy Line L(1, 7, 20, 175, 176) — The Soddy Line: Connects Incenter X₁ and Gergonne Point X₇ across the intouch triangle T_a T_b T_c and incircle. (The three mutually tangent vertex circles of radii s−a, s−b, s−c meet at T_a, T_b, T_c.)

2. Coordinate & Algebraic Law (RADICAL & PERSPECTION AXIS): Intersection with Euler Line at de Longchamps Point X_20 — Euler–Soddy Intersection: The Soddy line through X₁ and X₇ meets the Euler line {2, 3, 4, 20} at X₂₀. (Perpendicular to the Gergonne line (trilinear polar of X₇).)

3. Central Line Incidences (DE LONGCHAMPS INTERSECTION): Soddy Line {1, 7, 20} ∩ Euler Line {2, 3, 4, 20} — Crossing Central Axes: The Soddy line {1, 7, 20} and the Euler line {2, 3, 4, 20} intersect at X₂₀. (X₂₀ is the anticomplement of the orthocenter X₄ and reflection of X₄ in X₃.)

4. Collection 1 Network (SODDY CENTERS X₁₇₅ & X₁₇₆): X_1 • X_7 • X_20 — Soddy Collinearity: X₁, X₇, and X₂₀ anchor the Soddy axis on △ABC. (The inner and outer Soddy centers divide X₁ and X₇ harmonically with the Descartes curvatures.)

Collection 1 Centers Involved

X₁, X₇, \(X_{20}\), X₂, X₃, X₄