T_22 • Intouch (Contact) & Extouch Triangles

Category: Derived Central Triangles

The Intouch Triangle T_a T_b T_c formed by the points of tangency of the incircle (perspector X₇ Gergonne point) and the Extouch Triangle formed by the excircle touch points (perspector X₈ Nagel point).

Primary Construction & Central Lines

T_22 • Intouch (Contact) & Extouch Triangles Toggle Central Lines

Key Properties & Sequenced Operations

1. Definition (JOSEPH DIAZ GERGONNE (1818) & CHRISTIAN VON NAGEL (1836)): Intouch Triangle T_a T_b T_c (Perspector X_7) • Extouch Triangle E_a E_b E_c (Perspector X_8) — Contact Triangles: The intouch triangle T_a T_b T_c inscribed in the incircle has perspector X₇, while the extouch triangle has perspector X₈. (Intouch triangle is the pedal triangle of X₁ and cevian triangle of X₇.)

2. Coordinate & Algebraic Law (EQUAL AREA PROPERTY): Symmetric Touch Segments: BT_a = CE_a = s−b  •  CT_a = BE_a = s−c — Isotomic Touch Feet: Intouch perspector X₇ = 1/(s−a) : 1/(s−b) : 1/(s−c) and extouch perspector X₈ = s−a : s−b : s−c are reciprocal in barycentrics. (The intouch triangle T_a T_b T_c and extouch triangle E_a E_b E_c always have the exact same area: 2Δ²r / (abc).)

3. Central Line Incidences (COMPLEMENT OF GERGONNE POINT IS MITTENPUNKT X₉): Soddy Line {1, 7} • Nagel Line {1, 2, 8} • Mittenpunkt Line {2, 7, 9} — Contact Central Lines: Soddy axis {1, 7}, Nagel axis {1, 2, 8}, and Gergonne–Centroid–Mittenpunkt line {2, 7, 9}. (Under homothety at X₂, X₁ maps to X₈ (anticomplement), while X₉ is the complement of X₇.)

4. Collection 1 Network (SPLITTER SEGMENTS): X_7 ↔ X_8 • X_1 • X_2 • X_9 — Contact Quartet: Gergonne X₇ and Nagel X₈ pair across X₁, X₂, and X₉. (The cevians through X₈ bisect the perimeter of △ABC and are called the splitters of the triangle.)

Collection 1 Centers Involved

X₇, X₈, X₁, X₂, X₉, X₁₁