T_2 • Isotomic Conjugate

Category: Transformations & Conjugacies

The reflection of concurrent cevians across the side midpoints of triangle ABC, inverting barycentric coordinates u : v : w ↦ 1/u : 1/v : 1/w.

Primary Construction & Central Lines

T_2 • Isotomic Conjugate
< A B C >: Reference Triangle △ABC < M_a M_b M_c >: Medial Triangle △M_a M_b M_c < T_a T_b T_c >: Intouch (Contact) Triangle △T_a T_b T_c [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) [ A M_a ]: A-Median [ B M_b ]: B-Median [ C M_c ]: C-Median [ A T_a ]: Gergonne Cevian A–T_a [ B T_b ]: Gergonne Cevian B–T_b [ C T_c ]: Gergonne Cevian C–T_c incircle: Incircle of △ABC { 1, 7 }: Soddy Line { 2, 7 }: Centroid–Gergonne Line { 2, 8 }: Nagel Line A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) M_a = Midpoint M_a of BC (0.8061, 0.2999) M_b = Midpoint M_b of CA (0.3061, 0.2999) M_c = Midpoint M_c of AB (0.5000, 0.0000) T_a = Incircle Touchpoint T_a on BC (0.7673, 0.3599) T_b = Incircle Touchpoint T_b on CA (0.4082, 0.3999) T_c = Incircle Touchpoint T_c on AB (0.5714, 0.0000) X2 = CENTROID (0.5374, 0.2000) X8 = NAGEL POINT (0.4694, 0.1333) X1 = INCENTER (0.5714, 0.2333) X4 = ORTHOCENTER (0.6122, 0.3958) X7 = GERGONNE POINT (0.5903, 0.2769) A B C Ma Mb Mc Ta Tb Tc X2 X8 X1 X4 X7

Key Properties & Sequenced Operations

  1. Definition (MIDPOINT SYMMETRY • BD = CE): P = u : v : w  ⟷  P† = 1/u : 1/v : 1/w — Isotomic Conjugacy: Reflecting cevian feet across the side midpoints of the medial triangle M_a M_b M_c fixes the centroid X2 and swaps touch-point cevians of the incircle. (By Ceva’s theorem, reflecting cevian feet across M_a, M_b, M_c preserves concurrence.)

  2. Coordinate & Algebraic Law (AFFINE INVOLUTION): Barycentrics u : v : w ↦ 1/u : 1/v : 1/w — Reciprocal Barycentrics: Sub-area weights u : v : w invert to 1/u : 1/v : 1/w while trilinears map to (bc α)⁻¹ : (ca β)⁻¹ : (ab γ)⁻¹. (Isotomic conjugation commutes with every affine transformation of triangle ABC.)

  3. Central Line Incidences (TOUCH-POINT SYMMETRY): Gergonne–Nagel Isotomic Axis {7, 8} — Collection 1 Action: The Soddy line {1, 7, 8} and centroid axes {2, 7} and {2, 8} frame the Gergonne–Nagel isotomic pair. (Incircle and excircle tangency points lie at distance s − b and s − c from opposite vertices.)

  4. Collection 1 Network (ISOTOMIC PAIRS): X_2 Self-Isotomic • X_7 ↔ X_8 — Conjugate Network: X2 is self-isotomic, while Gergonne point X7 and Nagel point X8 are mutual isotomic conjugates. (Also pairs Incenter X1 ↔ X75 (isotomic conjugate of incenter) and Orthocenter X4 ↔ X69.)

Collection 1 Centers Involved

X2, X7, X8, X1, X4