F.37 Euler’s Inradius Parabola — Folios

Archival manuscript scans and diplomatic transcriptions from seelig’s notebooks for F.37 (Euler’s Inradius Parabola) (SeeligNotebooks).

Folio 0233 (Page 1 of 3): Euler’s Identity as a Downward Parabola r(d)

[Folio 0233]

Folio 0233 (original scan 0233.jpg) from resources/seelig/37_parabolas.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0233 (0233.jpg, Sheet 1 of 3):

Inverts \(d^2 = R^2 - 2rR\) into \(r(d) = \frac{R}{2} - \frac{d^2}{2R}\) and plots the parabola from vertex \((0, R/2)\) to roots \((\pm R, 0)\).

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

Editorial & Symbolic Notes

Archival scan 0233.jpg from Fred W. Seelig’s notebook dossier on F.37 Euler’s Inradius Parabola (Page 1 of 3). Companion OpenDocument worksheet(s): Incircle-Incenter, Circumcircle-Circumcenter. Classical Euclidean foundations: II.5, III.35. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(3) — Circumcenter.

Folio 0234 (Page 2 of 3): Factored Form & Power of the Incenter

[Folio 0234]

Folio 0234 (original scan 0234.jpg) from resources/seelig/37_parabolas.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0234 (0234.jpg, Sheet 2 of 3):

Rewrites \(2rR = (R-d)(R+d)\) as the chord-segment product of the incenter inside the circumcircle of radius \(R\).

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

Editorial & Symbolic Notes

Archival scan 0234.jpg from Fred W. Seelig’s notebook dossier on F.37 Euler’s Inradius Parabola (Page 2 of 3). Companion OpenDocument worksheet(s): Incircle-Incenter, Circumcircle-Circumcenter. Classical Euclidean foundations: II.5, III.35. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(3) — Circumcenter.

Folio 0235 (Page 3 of 3): Harmonic Division Along the Circumdiameter

[Folio 0235]

Folio 0235 (original scan 0235.jpg) from resources/seelig/37_parabolas.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0235 (0235.jpg, Sheet 3 of 3):

Relates the diameter segments \(R-d\) and \(R+d\) to the harmonic ratio \(k = (R+d)/(R-d)\).

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

Editorial & Symbolic Notes

Archival scan 0235.jpg from Fred W. Seelig’s notebook dossier on F.37 Euler’s Inradius Parabola (Page 3 of 3). Companion OpenDocument worksheet(s): Incircle-Incenter, Circumcircle-Circumcenter. Classical Euclidean foundations: II.5, III.35. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(3) — Circumcenter.