T_1 • Isogonal Conjugate

Category: Transformations & Conjugacies

The reflection of concurrent cevians across the interior angle bisectors of triangle ABC, inverting trilinear coordinates α : β : γ ↦ 1/α : 1/β : 1/γ.

Primary Construction & Central Lines

T_1 • Isogonal Conjugate Toggle Central Lines

Key Properties & Sequenced Operations

1. Definition (CEVIAN REFLECTION • ∠ACP = ∠BCP*): P = α : β : γ  ⟷  P* = 1/α : 1/β : 1/γ — Isogonal Conjugacy: Reflecting the cevians of X₂ across the angle bisectors through X₁ produces the symmedians concurring at X₆, while incircle and circumcircle anchor the metric duality. (Isogonal lines make equal angles with the interior angle bisectors at vertices A, B, and C.)

2. Coordinate & Algebraic Law (ALGEBRAIC INVOLUTION): (X_2, X_6) • (X_3, X_4) • (X_13, X_15) — Reciprocal Trilinears: α : β : γ maps to 1/α : 1/β : 1/γ and barycentrics u : v : w map to a²/u : b²/v : c²/w. (Points on the circumcircle map to the line at infinity; the sidelines of △ABC are singular.)

3. Central Line Incidences (ISOGONAL PAIRS): Classic Isogonal Pairs in Collection 1 — Collection 1 Action: Medians and symmedians pair {2, 6}, circumcenter and orthocenter pair {3, 4}, and Fermat and Isodynamic centers pair {13, 15}. (Self-isogonal fixed points are the incenter X₁ and the three excenters I_a, I_b, I_c.)

4. Collection 1 Network (GLOBAL DUALITY): X_1 Fixed • X_2 ↔ X_6 • X_3 ↔ X_4 • X_13 ↔ X_15 — Conjugate Network: X₁ is fixed; X₂ conjugates with X₆; X₃ conjugates with X₄; and X₁₃ conjugates with X₁₅. (Also pairs Gergonne X₇ ↔ X₅₅, Nagel X₈ ↔ X₅₇, and Napoleon X₁₇ ↔ X₁₈.)

Collection 1 Centers Involved

X₁, X₂, X₆, X₃, X₄, X₁₃, X₁₅