T_6 • The Euler Line

Category: Named Central Lines

Leonhard Euler’s 1765 central line passing through the Centroid X2, Circumcenter X3, Orthocenter X4, and Nine-Point Center X5 of every non-equilateral triangle.

Primary Construction & Central Lines

T_6 • The Euler Line
< A B C >: Reference Triangle △ABC < M_a M_b M_c >: Medial Triangle △M_a M_b M_c < H_a H_b H_c >: Orthic Triangle △H_a H_b H_c [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) [ A H_a ]: A-Altitude [ B H_b ]: B-Altitude [ C H_c ]: C-Altitude circumcircle: Circumcircle of △ABC nine_point_circle: Nine-Point Circle (Euler / Feuerbach Circle) { 2, 3 }: Euler 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) H_a = Altitude Foot H_a on BC (0.7053, 0.4559) H_b = Altitude Foot H_b on CA (0.5102, 0.4999) H_c = Altitude Foot H_c on AB (0.6122, 0.0000) X3 = CIRCUMCENTER (0.5000, 0.1021) X4 = ORTHOCENTER (0.6122, 0.3958) X5 = NINE-POINT CENTER (0.5561, 0.2489) X20 = DE LONGCHAMPS POINT (0.3878, -0.1916) X30 = EULER INFINITY POINT (0.7469, 0.7481) X2 = CENTROID (0.5374, 0.2000) A B C Ma Mb Mc Ha Hb Hc X3 X4 X5 X20 X30 X2

Key Properties & Sequenced Operations

  1. Definition (EULER (1765) • NOVI COMMENTARII): Euler Line L(2, 3, 4, 5, 20, 30) • |X_3 X_2| : |X_2 X_5| : |X_5 X_4| = 2 : 1 : 3 — The Euler Line: Connects X2, X3, X4, and X5 across the medial triangle M_a M_b M_c and nine-point circle. (On an equilateral triangle, X2, X3, X4, and X5 coincide and the Euler line is indeterminate.)

  2. Coordinate & Algebraic Law (EULER INFINITY POINT X30): Trilinear Equation: sin(2A) sin(B−C) α + sin(2B) sin(C−A) β + sin(2C) sin(A−B) γ = 0 — Shinagawa Coefficients: Every Euler center is a linear combination p·X4 + q·X3 with X2 = (1, 2) and X5 = (1, 1). (X30 is the point at infinity of the Euler line; X20 (de Longchamps) is the reflection of X4 in X3.)

  3. Central Line Incidences (2 : 1 : 3 SEGMENT RATIOS): Collinearity {2, 3, 4, 5, 20} — Central Line Spine: The Euler line {2, 3, 4, 5, 20} links the circumcenter, centroid, nine-point center, orthocenter, and de Longchamps point. (X2 trisects X3 X4, while X5 bisects X3 X4.)

  4. Collection 1 Network (COLLECTION 1 ANCHOR): X_3 • X_2 • X_5 • X_4 • X_20 — Ordered Euler Sequence: X2, X3, X4, X5, and X20 lie in exact rational ratio along the Euler axis. (Four of the first five centers in Kimberling’s Encyclopedia lie on the Euler line.)

Collection 1 Centers Involved

X2, X3, X4, X5, \(X_{20}\), \(X_{30}\)