T_15 • The Circumcircle¶
Category: Named Circles
The unique circle passing through vertices A, B, and C of triangle ABC, centered at the Circumcenter X3 with radius R = abc / (4Δ).
Primary Construction & Central Lines¶
Key Properties & Sequenced Operations¶
Definition (EUCLID BOOK IV • PROPOSITION 5):
Circumcircle (Center X_3, Radius R = abc / 4Δ) • 2R = a/sin A = b/sin B = c/sin C— The Circumcircle: Passes through A, B, C with center X3 on the perpendicular bisectors of △ABC. (Barycentric equation: a²vw + b²wu + c²uv = 0; trilinear equation: aβγ + bγα + cαβ = 0.)Coordinate & Algebraic Law (SIMSON LINE LOCUS):
Isogonal Transform of the Line at Infinity (u + v + w = 0)— Projective & Metric Role: X3 = cos A : cos B : cos C anchors the circumcircle of radius R = abc / (4Δ). (The pedal feet of a point P are collinear (forming a Simson line) if and only if P lies on the circumcircle.)Central Line Incidences (RADIAL INVERSION CENTER):
Euler Line {2, 3, 4, 5} & Brocard Axis {3, 6, 15, 16}— Orthogonal Axes at X3: X3 is the origin of the Euler line {2, 3, 4, 5} and the Brocard axis {3, 6, 15, 16}. (X15 and X16 are mutual inverses in the circumcircle along the Brocard axis.)Collection 1 Network (TANGENTIAL POLARITY):
X_3 • X_2 • X_4 • X_5 • X_6 • X_15 • X_16— Circumcircle Network: X3 links X2, X4, X5 on the Euler line and X6, X15, X16 on the Brocard axis. (The tangents to the circumcircle at A, B, C form the tangential triangle, whose perspector is X6.)