T_4 • Harmonic Conjugate & Cross-Ratio

Category: Transformations & Conjugacies

Four collinear centers (A, B; C, D) form a harmonic range when C and D divide segment AB internally and externally in the same ratio, yielding cross-ratio −1.

Primary Construction & Central Lines

T_4 • Harmonic Conjugate & Cross-Ratio
< 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) incircle: Incircle of △ABC nine_point_circle: Nine-Point Circle (Euler / Feuerbach Circle) { 2, 3 }: Euler Line { 1, 5 }: Feuerbach 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) X2 = CENTROID (0.5374, 0.2000) X3 = CIRCUMCENTER (0.5000, 0.1021) X4 = ORTHOCENTER (0.6122, 0.3958) X1 = INCENTER (0.5714, 0.2333) X11 = FEUERBACH POINT (0.7347, 0.0667) X12 = {X1,X5}-HARMONIC CONJUGATE OF X11 (0.5641, 0.2407) X5 = NINE-POINT CENTER (0.5561, 0.2489) A B C Ma Mb Mc X2 X3 X4 X1 X11 X12 X5

Key Properties & Sequenced Operations

  1. Definition (PROJECTIVE INVARIANCE): (A, B; C, D) = (AC / CB) : (AD / DB) = −1 — Harmonic Range: On the Euler line through X5, internal and external division of X3 and X4 with X2 links the medial triangle M_a M_b M_c and nine-point circle. (Cross-ratio (A, B; C, D) = −1 is preserved under every projective collineation.)

  2. Coordinate & Algebraic Law (HARMONIC MEAN): Euler Range (X_3, X_4; X_2, X_5) = −1 — Homogeneous Linear Combination: If C = p·A + q·B, then its harmonic conjugate with respect to A and B is D = p·A − q·B, satisfying 2/AB = 1/AC + 1/AD. (On the Euler line: X2 : X5 divide X3 X4 in ratios 1 : 2 and 1 : −1.)

  3. Central Line Incidences (SIMILICENTER DUALITY): Euler Line {2, 3, 4, 5} & Feuerbach Line {1, 5, 11, 12} — Collection 1 Harmonic Axes: The Euler line {2, 3, 4, 5} and Feuerbach line {1, 5, 11, 12} meet at the Nine-Point Center X5. (Internal and external centers of similitude of any two circles are harmonic conjugates with respect to their centers.)

  4. Collection 1 Network (APOLLONIAN HARMONICS): (X_2, X_4; X_3, X_5) = −1 • (X_1, X_5; X_11, X_12) = −1 — Harmonic Quartets: Euler centers X2, X3, X4, X5 and Feuerbach centers X1, X5, X11, X12. (On the Brocard axis, (X3, X6; X15, X16) also forms a classical harmonic structure.)

Collection 1 Centers Involved

X2, X3, X4, X5, X1, X11, X12