T_18 • The Apollonian Circles

Category: Named Circles

The three circles of Apollonius passing through each vertex and its interior and exterior angle-bisector feet, intersecting at the 1st and 2nd Isodynamic Points X15 and X16.

Primary Construction & Central Lines

T_18 • The Apollonian Circles
< 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 apollonian_circle: Apollonian circle { 3, 6 }: Brocard Axis { 6, 13 }: Central Line L(6, 13) { 6, 14 }: Central Line L(6, 14) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X16 = 2nd ISODYNAMIC POINT (1.8453, 2.3239) X3 = CIRCUMCENTER (0.5000, 0.1021) X6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X13 = 1st ISOGONIC CENTER (FERMAT POINT, TORRICELLI POINT) (0.5879, 0.2819) X14 = 2nd ISOGONIC CENTER (0.9134, -0.1143) O_C = Kosnita Circumcenter O_C (3.2727, 0.0000) X15 = 1st ISODYNAMIC POINT (0.5519, 0.1878) A B C X16 X3 X6 X13 X14 OC X15

Key Properties & Sequenced Operations

  1. Definition (APOLLONIUS OF PERGA • LOCUS OF CONSTANT RATIO): Three Apollonian Circles: PB/PC = c/b • PC/PA = a/c • PA/PB = b/a — The Apollonian Circles: Intersect on the Brocard axis at the 1st Isodynamic Point X15 and 2nd Isodynamic Point X16, orthogonal to the circumcircle. (Diameter of the A-Apollonian circle is the harmonic segment between interior and exterior bisector feet on BC.)

  2. Coordinate & Algebraic Law (EQUILATERAL PEDAL TRIANGLES): Coaxial Pencil Centered on the Lemoine Axis • Radical Axis = Brocard Axis — Equidistant Pedal Property: At X15 and X16, vertex distances satisfy a·|PA| = b·|PB| = c·|PC| and the pedal triangle is equilateral. (X15 and X16 are the unique points in the plane whose pedal triangles on △ABC are equilateral.)

  3. Central Line Incidences (RADICAL CHORD): Brocard Axis {3, 6, 15, 16} & Symmedian Crossings {6, 13, 16} • {6, 14, 15} — Apollonian Axes: Radical axis {3, 6, 15, 16} meets Fermat–Isodynamic lines {6, 13, 16} and {6, 14, 15} at X6. (All three Apollonian circles intersect orthogonally with the circumcircle of △ABC.)

  4. Collection 1 Network (MOBIUS EQUILATERAL INVARIANCE): X_15 • X_16 • X_3 • X_6 • X_13 • X_14 — Isodynamic Network: X15 and X16 pair isogonally with Fermat points X13 and X14 across X3, X6. (A Möbius transformation sending X15 to 0 and X16 to ∞ maps △ABC to an equilateral triangle.)

Collection 1 Centers Involved

X15, X16, X3, X6, X13, X14