F.11 Circumcenter

Perpendicular Bisectors, Cartesian Determinant Formulas, Circumcircle Subtriangles, and the Tangential Triangle

The perpendicular bisectors of the three sides of triangle ABC concur at the circumcenter O = X(3), equidistant from A, B, C by the circumradius R = abc/(4K), with homogeneous barycentric coordinates a^2 S_A : b^2 S_B : c^2 S_C.

Interactive GEOMETOR construction for F.11 (Circumcenter). Hover any geometric label in the text or figure to highlight the corresponding element.

A : Point B : Point C : Point ===== M_A = midpoint(B, C) M_B = midpoint(C, A) M_C = midpoint(A, B) perp_a = perpendicular(M_A, line(B, C)) perp_b = perpendicular(M_B, line(C, A)) O = intersect(perp_a, perp_b) circumcircle = circle(O, A)

Mathematical Synthesis

Dated May 25 and May 31, 2011 (SeeligNotebooks), Seelig’s study of the circumcenter O \(= X(3)\) derives the exact Cartesian coordinates \((x_0, y_0)\) from vertex coordinates (correcting a sign error in Joseph O’Rourke’s published formula), transcribes Christopher Morley’s poem ‘A circle is a happy thing to be’, and decomposes \(\triangle ABC\) into the three isosceles circumcircle subtriangles \(\triangle AOB, \triangle BOC, \triangle COA\) of altitudes \(h_C = \sqrt{R^2 - (c/2)^2}\).

In Folios 5–6, Seelig constructs the tangential triangle bounded by the tangent lines to the circumcircle at A, B, C, contrasting the acute case (where the tangential triangle encloses \(\triangle ABC\)) with the obtuse case. On the canonical \((5, 6, 7)\) triangle, \(R = \frac{35\sqrt{6}}{24} \approx 3.57217\) and \(O = (7/2, 5\sqrt{6}/18) \approx (3.5, 0.6804)\).

Key Algebraic Identities

\[R = \frac{a b c}{4 K}, \qquad O = \left(a^2 S_A : b^2 S_B : c^2 S_C\right) = \left(\sin 2\alpha : \sin 2\beta : \sin 2\gamma\right)\]

\[D = 2\left[x_A(y_C - y_B) + x_B(y_A - y_C) + x_C(y_B - y_A)\right], \qquad x_0 = \frac{\sum (x_A^2 + y_A^2)(y_C - y_B)}{D}\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.11 Circumcenter — Folios (Folios 0066–0071, 6 scanned manuscript pages in resources/seelig/11_circumcenter)
Euclid’s Elements	III.1, IV.5 (EuclidElements)
GEOMETOR Elements (G-Index)	G.3 extremities of a segment, G.4 segment
Encyclopedia of Triangle Centers	X(3) — Circumcenter, X(26) — Circumcenter of the Tangential Triangle (Kimberling1998)
Seelig Spreadsheets	Circumcircle-Circumcenter, Circumcircle Solver, General Reference (Spreadsheets)

Constructive Dependency Graph

Prerequisites: F.3 Biquadratic Forms in Heron’s Formula

Required for: F.4 Carnot’s Theorem, F.13 Compare Triangles in Circles, F.20 Extangent, F.28 Incenter, F.32 Markowski’s Theorem, F.34 Nine-Point Circle, F.37 Euler’s Inradius Parabola, F.52 Homogeneous Barycentric Coordinates