T_19 • The Brocard & Orthocentroidal Circles

Category: Named Circles

Two fundamental diameter circles of triangle geometry: the Brocard Circle on diameter X3 X6 and the Orthocentroidal Circle on diameter X2 X4.

Primary Construction & Central Lines

T_19 • The Brocard & Orthocentroidal Circles
< 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) brocard_circle: Brocard Circle (Seven-Point Circle) orthocentroidal_circle: Orthocentroidal Circle { 3, 6 }: Brocard Axis { 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) X2 = CENTROID (0.5374, 0.2000) X3 = CIRCUMCENTER (0.5000, 0.1021) X4 = ORTHOCENTER (0.6122, 0.3958) X5 = NINE-POINT CENTER (0.5561, 0.2489) X19 = CLAWSON POINT (0.6246, 0.4244) X182 = MIDPOINT OF BROCARD DIAMETER (0.5500, 0.1846) X381 = MIDPOINT OF X2 AND X4 (0.5748, 0.2979) X6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) A B C Ma Mb Mc X2 X3 X4 X5 X19 X182 X381 X6

Key Properties & Sequenced Operations

  1. Definition (BROCARD (1881) & GUINAND (1984)): Brocard Circle (Diameter X_3 X_6) • Orthocentroidal Circle (Diameter X_2 X_4) — Diameter Circles: The Brocard circle on diameter X3 X6 and the orthocentroidal circle on diameter X2 X4. (Every pedal of X6 onto the perpendicular bisectors lies on the Brocard circle.)

  2. Coordinate & Algebraic Law (GUINAND’S THEOREM): Brocard Circle Center X_182 (Midpoint of X_3 X_6) • Orthocentroidal Center X_381 (Midpoint of X_2 X_4) — Midpoint Centers: Brocard diameter X3 X6 and Euler diameter X2 X4 bound the interior locus of X1 and X6. (In any non-equilateral triangle, X1 lies inside the orthocentroidal disk and never on the Euler line unless isosceles.)

  3. Central Line Incidences (RADICAL AXIS): Brocard Axis {3, 6} & Euler Line {2, 3, 4, 5} — Diameter Axes: The Brocard axis {3, 6} and Euler line {2, 3, 4, 5} form the perpendicular diameters of the two circles at X3. (The radical axis of the circumcircle and Brocard circle is the Lemoine line.)

  4. Collection 1 Network (CLAWSON & SYMMEDIAN GEOMETRY): X_2 • X_3 • X_4 • X_5 • X_6 — Five Classical Centers: X2, X3, X4, X5 on the Euler line and X6 on the Brocard circle. (The orthocentroidal and Brocard circles are linked by pedal projections and isogonal inversion.)

Collection 1 Centers Involved

X2, X3, X4, X5, X6, X19