F.20 Extangent

The Extangents Triangle, Excircles, and Differentiation of Euler’s Incenter-Circumcenter Formula

The external common tangents to the incircle and circumcircle (or excircles) bound the extangents triangle, while Euler’s relation d_{IO}^2 = R^2 - 2rR has derivative d(d_{IO})/dR = (R - r)/sqrt(R^2 - 2rR).

Interactive GEOMETOR construction for F.20 (Extangent). 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) I_a = excenter_a(A, B, C) I_b = excenter_b(A, B, C) I_c = excenter_c(A, B, C) incircle = circle(I, T_C) excircle_C = circle(I_c, E_C)

Mathematical Synthesis

In Topic 20 (SeeligNotebooks), Seelig constructs the extangents triangle formed by the external tangents to the incenter / excircle configuration and differentiates Euler’s distance formula \(d_{IO} = \sqrt{R^2 - 2rR}\) with respect to \(R\), confirming Euler’s inequality \(R \ge 2r\) with equality only for the equilateral triangle.

Key Algebraic Identities

\[d_{IO}^2 = R^2 - 2 r R, \qquad \frac{d}{d R}(d_{IO}) = \frac{R - r}{\sqrt{R^2 - 2 r R}}, \qquad r_a = \frac{K}{s - a}, \quad r_b = \frac{K}{s - b}, \quad r_c = \frac{K}{s - c}\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.20 Extangent — Folios (Folios 0127–0127, 1 scanned manuscript page in resources/seelig/20_extangent)
Euclid’s Elements	IV.4 (EuclidElements)
GEOMETOR Elements (G-Index)	G.3 extremities of a segment
Encyclopedia of Triangle Centers	X(1) — Incenter, X(3) — Circumcenter, X(40) — Bevan Point, X(55) — Internal Center of Similitude of Incircle and Circumcircle, X(56) — External Center of Similitude of Incircle and Circumcircle (Kimberling1998)
Seelig Spreadsheets	Excenters, Incircle-Incenter, Centers of Similitude (Spreadsheets)

Constructive Dependency Graph

Prerequisites: F.5 Centers of Similitude, F.11 Circumcenter, F.28 Incenter