F.37 Euler’s Inradius Parabola

Parabolic Locus r(d) = (R^2 - d^2)/(2R) and Harmonic Factorization of Euler’s Identity

For a fixed circumradius R, Euler’s relation d^2 = R^2 - 2 r R defines a downward-opening parabola r(d) = R/2 - d^2/(2R) = (R - d)(R + d)/(2R) with vertex (0, R/2) at the equilateral triangle and roots at the degenerate limits d = +/- R.

Interactive GEOMETOR construction for F.37 (Euler’s Inradius Parabola). Hover any geometric label in the text or figure to highlight the corresponding element.

V : Point F : Point directrix : Line P : Point ===== Q = foot(P, directrix) s_fp = segment(F, P) s_pq = segment(P, Q) tangent = bisector(F, P, Q)

Mathematical Synthesis

In this three-folio study (SeeligNotebooks), Seelig reinterprets Euler’s triangle distance theorem \(d^2 = R^2 - 2rR\) (where \(d = |IO|\)) as a functional locus \(r(d) = \frac{R}{2} - \frac{d^2}{2R}\) over a fixed circumcircle of radius \(R\). Plotting \(r\) against \(d\) yields a downward-opening parabola whose vertex \((0, R/2)\) represents the equilateral triangle (\(r = R/2\), Euler’s inequality maximum) and whose horizontal intercepts \(d = \pm R\) represent degenerate flat triangles (\(r = 0\)).

Factoring the numerator as \(r(d) = \frac{(R-d)(R+d)}{2R}\), Seelig connects the segments \(R-d\) and \(R+d\) partitioned by the incenter I along a diameter of the circumcircle to the power of a point \((R-d)(R+d) = 2rR\) and harmonic conjugate division.

Key Algebraic Identities

\[d^2 = R^2 - 2 r R \iff r(d) = \frac{R}{2} - \frac{d^2}{2 R} = \frac{(R - d)(R + d)}{2 R}\]

\[r_{\max} = r(0) = \frac{R}{2}, \qquad (R - d)(R + d) = 2 r R\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.37 Euler’s Inradius Parabola — Folios (Folios 0233–0235, 3 scanned manuscript pages in resources/seelig/37_parabolas)
Euclid’s Elements	II.5, III.35 (EuclidElements)
GEOMETOR Elements (G-Index)	G.7 plane surface, G.8 boundary
Encyclopedia of Triangle Centers	X(1) — Incenter, X(3) — Circumcenter (Kimberling1998)
Seelig Spreadsheets	Incircle-Incenter, Circumcircle-Circumcenter (Spreadsheets)

Constructive Dependency Graph

Prerequisites: F.1 Albert’s Parabola, F.11 Circumcenter, F.29 Inradius

Required for: F.41 Geometric Problems and Extremal Loci