T_5 • Circle Inversion

Category: Transformations & Conjugacies

Radial mapping across a reference circle of center O and radius R sending point P to P’ on ray OP such that OP · OP’ = R².

Primary Construction & Central Lines

T_5 • Circle Inversion
< 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) incircle: Incircle of △ABC circumcircle: Circumcircle of △ABC { 3, 6 }: Brocard Axis { 3, 4 }: Euler Line 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) X6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X16 = 2nd ISODYNAMIC POINT (1.8453, 2.3239) X1 = INCENTER (0.5714, 0.2333) X4 = ORTHOCENTER (0.6122, 0.3958) X15 = 1st ISODYNAMIC POINT (0.5519, 0.1878) A B C X3 X6 X16 X1 X4 X15

Key Properties & Sequenced Operations

  1. Definition (ORTHOGONAL CIRCLES): OP · OP' = R²  •  X_15 ↔ X_16 in Circumcircle — Circumcircle Inversion: Radial inversion across the circumcircle centered at X3 exchanges the 1st Isodynamic point X15 and 2nd Isodynamic point X16. (Any circle passing through a pair of inverse points is strictly orthogonal to the circle of inversion.)

  2. Coordinate & Algebraic Law (CONFORMAL INVOLUTION): Radial Product |X_3 X_15| · |X_3 X_16| = R² — Algebraic Formula: Inversion of P in the circumcircle reflects along ray from X3 with radial product OP · OP’ = R². (Circle inversion preserves angles between curves and maps generalized circles to generalized circles.)

  3. Central Line Incidences (RADIAL AXES THROUGH X3): Brocard Axis {3, 6, 15, 16} & Euler Line {3, 4} — Collection 1 Inversion Axes: The Brocard axis {3, 6, 15, 16} carries the inverse pair X15, X16 while {3, 4} carries the orthocenter inverse X186. (Every line through the circumcenter X3 is invariant as a set under circumcircle inversion.)

  4. Collection 1 Network (INVERSION CATALOG): X_15 ↔ X_16 on Brocard Axis {3, 6, 15, 16} — Inverse Partners: On the Brocard axis through X3 and X6, X15 and X16 are exact circumcircle inverses. (Incenter X1 inverts in the circumcircle to X36; Orthocenter X4 inverts to X186.)

Collection 1 Centers Involved

X3, X6, X15, X16, X1, X4