T_20 • Medial & Anticomplementary Triangles

Category: Derived Central Triangles

The Medial Triangle M_a M_b M_c formed by the side midpoints (cevian triangle of X2) and the Anticomplementary Triangle formed by lines through A, B, C parallel to the opposite sides.

Primary Construction & Central Lines

T_20 • Medial & Anticomplementary Triangles
< A B C >: Reference Triangle △ABC < M_a M_b M_c >: Medial Triangle △M_a M_b M_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 nine_point_circle: Nine-Point Circle (Euler / Feuerbach Circle) spieker_circle: Spieker Circle { 2, 3 }: Euler Line { 1, 2 }: 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) X1 = INCENTER (0.5714, 0.2333) X3 = CIRCUMCENTER (0.5000, 0.1021) X4 = ORTHOCENTER (0.6122, 0.3958) X5 = NINE-POINT CENTER (0.5561, 0.2489) X8 = NAGEL POINT (0.4694, 0.1333) X10 = SPIEKER CENTER (0.5204, 0.1833) X2 = CENTROID (0.5374, 0.2000) A B C Ma Mb Mc X1 X3 X4 X5 X8 X10 X2

Key Properties & Sequenced Operations

  1. Definition (CEVIAN & ANTICEVIAN TRIANGLES OF X2): Medial Triangle M_a M_b M_c (Scale −1/2) • Anticomplementary Triangle (Scale −2) — Medial & Anticomplementary Hierarchy: Centered at X2, the medial triangle M_a M_b M_c has circumcircle equal to the nine-point circle and incircle equal to the Spieker circle. (Area(Medial) = 1/4 Area(△ABC); Area(Anticomplementary) = 4 Area(△ABC).)

  2. Coordinate & Algebraic Law (THREE-GENERATION LADDER): Center Mapping: X_4 (Anticomp) ↦ X_3 (△ABC) ↦ X_5 (Medial)  •  X_8 ↦ X_1 ↦ X_10 — Shift of Centers: Passing from anticomplementary to △ABC to medial maps X4 ↦ X3 ↦ X5 and X8 ↦ X1 ↦ X10. (△ABC is simultaneously the medial triangle of its anticomplementary triangle.)

  3. Central Line Incidences (SHARED CENTROID X2): Euler Line {2, 3, 4, 5} & Nagel Line {1, 2, 8, 10} — Dual Homothetic Axes: The Euler line {2, 3, 4, 5} and Nagel line {1, 2, 8, 10} cross at the common centroid X2. (The medial, reference, and anticomplementary triangles all share the exact same centroid X2.)

  4. Collection 1 Network (UNIVERSAL TEMPLATE): X_2 • (X_4, X_3, X_5) • (X_8, X_1, X_10) — Homothetic Triads: X2 governs the Euler triad X3, X4, X5 and the Nagel triad X1, X8, X10. (Every triangle center Xn has a medial counterpart (complement) and anticomplementary counterpart (anticomplement).)

Collection 1 Centers Involved

X2, X1, X3, X4, X5, X8, X10