T_8 • The Brocard Axis

Category: Named Central Lines

The central line passing through the Circumcenter X3, Symmedian Point X6, and the 1st and 2nd Isodynamic Points X15 and X16, perpendicular to the Lemoine axis.

Primary Construction & Central Lines

T_8 • The Brocard Axis
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) circumcircle: Circumcircle of △ABC brocard_circle: Brocard Circle (Seven-Point Circle) { 3, 6 }: Brocard Axis A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X3 = CIRCUMCENTER (0.5000, 0.1021) X15 = 1st ISODYNAMIC POINT (0.5519, 0.1878) X16 = 2nd ISODYNAMIC POINT (1.8453, 2.3239) X182 = MIDPOINT OF BROCARD DIAMETER (0.5500, 0.1846) X6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) A B C X3 X15 X16 X182 X6

Key Properties & Sequenced Operations

  1. Definition (HENRI BROCARD & ÉMILE LEMOINE): Brocard Axis L(3, 6, 15, 16) ⊥ Lemoine Axis — The Brocard Axis: Connects Circumcenter X3 and Symmedian Point X6 with Isodynamic Points X15, X16 across the circumcircle. (Isogonal conjugate of the Brocard axis is the Kiepert hyperbola.)

  2. Coordinate & Algebraic Law (ISOGONAL TRANSFORM TO KIEPERT HYPERBOLA): Trilinear Equation: bc(b²−c²)α + ca(c²−a²)β + ab(a²−b²)γ = 0 — Radical Chord of Apollonian Circles: X3 = cos A : cos B : cos C and X6 = a : b : c span the axis containing X15, X16. (Every point on the Kiepert hyperbola has its isogonal conjugate on the Brocard axis.)

  3. Central Line Incidences (DIAMETER OF BROCARD CIRCLE): Collinearity {3, 6, 15, 16} — Brocard Collinearity: The Brocard axis {3, 6, 15, 16} is the common radical line of the three Apollonian circles. (Segment X3 X6 is the diameter of the Brocard circle passing through the two Brocard points.)

  4. Collection 1 Network (KIEPERT DUALITY): X_3 • X_6 • X_15 • X_16 — Brocard Quartet: X3, X6, X15, and X16 align on the Brocard axis. (Isogonal conjugation maps the Kiepert centers {X2, X4, X13, X14} onto the Brocard axis {X6, X3, X15, X16}.)

Collection 1 Centers Involved

X3, X6, X15, X16