T_10 • The OI Line (Incenter–Circumcenter Axis)¶
Category: Named Central Lines
The central line joining the Incenter X1 and Circumcenter X3, carrying the Bevan Point X40 and the insimilicenter X55 and exsimilicenter X56 of the incircle and circumcircle.
Primary Construction & Central Lines¶
Key Properties & Sequenced Operations¶
Definition (EULER’S INEQUALITY R ≥ 2r):
OI Line L(1, 3, 40, 55, 56, 57) • |X_1 X_3|² = R(R − 2r)— The OI Line: Connects Incenter X1 and Circumcenter X3 as the line of centers of the incircle and circumcircle. (Since |X1 X3|² = R(R − 2r) ≥ 0, the circumradius R is at least twice the inradius r.)Coordinate & Algebraic Law (PONCELET PORISM AXIS):
Similicenters X_55 (Internal) & X_56 (External) • Bevan Point X_40— Similitude & Reflection: X3 is the midpoint of X1 and Bevan point X40; X55, X56 divide X1 X3 in ratio r : R. (The OI line is the axis of symmetry for Poncelet’s bicentric triangle family between the incircle and circumcircle.)Central Line Incidences (CIRCUMCIRCLE–INCIRCLE AXIS):
Collinearity {1, 3, 40, 55, 56}— OI Central Axis: The line {1, 3, 40, 55, 56} governs the metric separation between inscribed and circumscribed circles. (Contains X1, X3, X36 (circumcircle inverse of X1), X40, X55, X56, and X57.)Collection 1 Network (ISOGONAL LINKS TO X7 & X8):
X_1 • X_3 • X_40 • X_55 • X_56— OI Line Centers: X1, X3, X40, X55, and X56 align along the OI axis. (X55 is the isogonal conjugate of Gergonne X7; X56 is the isogonal conjugate of X55’s harmonic partner.)