F.29 Inradius

Special Inradius Values, Conway Intouch Segments, and the Gergonne Point

The inradius r = K / s = (1/2) sqrt(((b+c-a)(c+a-b)(a+b-c))/(a+b+c)) = 4 R sin(alpha/2) sin(beta/2) sin(gamma/2) takes integer values on primitive Pythagorean triples and partitions the perimeter into Conway segments x, y, z whose touchpoints T_A, T_B, T_C concur at the Gergonne point Ge = X(7).

Interactive GEOMETOR construction for F.29 (Inradius). Hover any geometric label in the text or figure to highlight the corresponding element.

A : Point B : Point C : Point ===== I = incenter(A, B, C) T_A = foot(I, line(B, C)) T_B = foot(I, line(C, A)) T_C = foot(I, line(A, B)) Ge = intersection(line(A, T_A), line(B, T_B)) c_in = circle(I, T_A)

Mathematical Synthesis

Seelig’s three-folio dossier on the inradius (SeeligNotebooks) opens with his master reference sheet :term:`Inradius <incenter>` / Special Inradius Values, pairing the Heron radical \(r = \frac{1}{2}\sqrt{\frac{(b+c-a)(c+a-b)(a+b-c)}{a+b+c}} = \frac{K}{s}\) and the trigonometric product \(r = 4R\sin\frac{\alpha}{2}\sin\frac{\beta}{2}\sin\frac{\gamma}{2}\) with a five-row benchmark table: Pythagorean triples \((3,4,5)\mapsto r=1\), \((5,12,13)\mapsto r=2\), \((8,15,17)\mapsto r=3\), and \((7,24,25)\mapsto r=3\); the \(30^\circ\text{--}60^\circ\text{--}90^\circ\) triangle; and the golden triangle \((\phi b, \phi b, b)\) with \(r = \frac{b}{2}\sqrt{5-2\sqrt{5}}\).

In Folios 2–3, Seelig connects Conway triangle notation (\(x = s-a, y = s-b, z = s-c\)) to the incircle contact points TA, TB, TC and their concurrency at the Gergonne point Ge (X(7) — Gergonne Point), verifying \(r = \sqrt{xyz/(x+y+z)}\) on a measured scalene drawing (\(a=28.3, b=19, c=30.8, s=39, K=263\)) and on the canonical \((5,6,7)\) triangle (\(x=4, y=3, z=2, s=9, r = \frac{2\sqrt{6}}{3}\)).

Key Algebraic Identities

\[r = \frac{K}{s} = \frac{1}{2}\sqrt{\frac{(b+c-a)(c+a-b)(a+b-c)}{a+b+c}} = \sqrt{\frac{x y z}{x + y + z}}\]

\[r = 4 R \sin\frac{\alpha}{2}\sin\frac{\beta}{2}\sin\frac{\gamma}{2}, \qquad x = s - a,\; y = s - b,\; z = s - c\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.29 Inradius — Folios (Folios 0207–0209, 3 scanned manuscript pages in resources/seelig/29_inradius)
Euclid’s Elements	IV.4 (EuclidElements)
GEOMETOR Elements (G-Index)	G.8 boundary
Encyclopedia of Triangle Centers	X(1) — Incenter, X(7) — Gergonne Point (Kimberling1998)
Seelig Spreadsheets	Incircle-Incenter, General Reference (Spreadsheets)

Constructive Dependency Graph

Prerequisites: F.14 Conway Notation, F.28 Incenter

Required for: F.32 Markowski’s Theorem, F.33 Euclidean (m, n) Generators, F.37 Euler’s Inradius Parabola, F.42 Altitude of a Pythagorean Triple Triangle, F.48 Verisimilitude, Corner Subtriangles, and Excircle Homothety, F.55 Compass Constructions and Angle-to-Side Triangle Reconstruction