F.28 Incenter

Concurrency of Angle Bisectors, Barycentric Coordinates a:b:c, Contact Vectors, and Incenter-Origin Shift

The three interior angle bisectors of triangle ABC concur at the incenter I = X(1) with homogeneous barycentric coordinates a : b : c, inradius r = K/s satisfying 1/r = 1/h_a + 1/h_b + 1/h_c, and Euler’s distance d_{IO}^2 = R(R - 2r).

Interactive GEOMETOR construction for F.28 (Incenter). Hover any geometric label in the text or figure to highlight the corresponding element.

A : Point B : Point C : Point ===== bis_A = angle_bisector(B, A, C) bis_B = angle_bisector(C, B, A) I = intersect(bis_A, bis_B) T_C = foot(I, line(A, B)) incircle = circle(I, T_C)

Mathematical Synthesis

Dated June 1 and November 8, 2011 and citing Nathan Altshiller-Court’s College Geometry (AltshillerCourt2007, SeeligNotebooks), Topic 28 develops the complete coordinate, vector, and metric theory of the incenter I \(= X(1)\).

Seelig records the Cartesian weighted average \(\vec{I} = (a\vec{A} + b\vec{B} + c\vec{C})/(2s)\), the vector touchpoints TA, TB, TC along the sides using reduced lengths \(x = s-a, y = s-b, z = s-c\), the reciprocal altitude identity \(1/r = 1/h_a + 1/h_b + 1/h_c\), Euler’s relation \(d^2 = R(R - 2r)\), and a coordinate translation placing the incenter at the origin \((0, 0)\). On the canonical \((5, 6, 7)\) triangle, \(r = \frac{2\sqrt{6}}{3} \approx 1.6330\) and \(I = (4, \frac{2\sqrt{6}}{3})\).

Key Algebraic Identities

\[\vec{I} = \frac{a \vec{A} + b \vec{B} + c \vec{C}}{a + b + c}, \qquad r = \frac{K}{s} = \sqrt{\frac{(s - a)(s - b)(s - c)}{s}}, \qquad \frac{1}{r} = \frac{1}{h_a} + \frac{1}{h_b} + \frac{1}{h_c}\]

\[2 r R = \frac{a b c}{a + b + c}, \qquad d_{IO}^2 = R(R - 2 r) \implies R \ge 2 r\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.28 Incenter — Folios (Folios 0202–0206, 5 scanned manuscript pages in resources/seelig/28_incenter)
Euclid’s Elements	IV.4, VI.3 (EuclidElements)
GEOMETOR Elements (G-Index)	G.3 extremities of a segment
Encyclopedia of Triangle Centers	X(1) — Incenter, X(7) — Gergonne Point (Kimberling1998)
Seelig Spreadsheets	Incircle-Incenter, General Reference, 5-6-7 A,B,C (Spreadsheets)

Constructive Dependency Graph

Prerequisites: F.3 Biquadratic Forms in Heron’s Formula, F.11 Circumcenter

Required for: F.4 Carnot’s Theorem, F.19 Equal Incircles Theorem, F.20 Extangent, F.29 Inradius, F.52 Homogeneous Barycentric Coordinates