F.32 Markowski’s Theorem — Folios

Archival manuscript scans and diplomatic transcriptions from seelig’s notebooks for F.32 (Markowski’s Theorem) (SeeligNotebooks).

Folio 0221 (Page 1 of 2): Markowski’s Theorem & Midpoint Inversion (FWS 4/23/2015)

[Folio 0221]

Folio 0221 (original scan 0221.jpg) from resources/seelig/32_markowski.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0221 (0221.jpg, Sheet 1 of 2):

Combines \(d_{IO} = \sqrt{R^2-2rR}\) and \(JKL = \frac{1}{2}r^2 R\) to derive \(JKL = \frac{(R^2-d_{IO}^2)^2}{8R}\), and records the midpoint inversion circle radius \(\rho_I = \sqrt{R^2-(a/2)^2}\).

\[J K L = \frac{1}{2} r^2 R = \frac{(R^2 - d_{IO}^2)^2}{8 R}, \qquad d_{IO}^2 = R^2 - 2 r R\]

Editorial & Symbolic Notes

Archival scan 0221.jpg from Fred W. Seelig’s notebook dossier on F.32 Markowski’s Theorem (Page 1 of 2). Companion OpenDocument worksheet(s): Circumcircle-Circumcenter, Incircle-Incenter. Classical Euclidean foundations: III.35, IV.5. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(3) — Circumcenter.

Folio 0222 (Page 2 of 2): Equilateral & 30-60-90 Triangle Verifications (FWS 4/23/2015)

[Folio 0222]

Folio 0222 (original scan 0222.jpg) from resources/seelig/32_markowski.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0222 (0222.jpg, Sheet 2 of 2):

Verifies \(JKL = R^3/8\) for the equilateral triangle and \(JKL = (2-\sqrt{3})/4\) for the \(30^\circ\text{--}60^\circ\text{--}90^\circ\) triangle (\(R=1\)), where an extra squaring of \(d_{IO}^2 = 2-\sqrt{3}\) prompted the marginal note ‘Stumped! FWS 4/23/2015’.

\[J + K + L = 2 R + r, \qquad \rho_I = \sqrt{R^2 - \left(\frac{a}{2}\right)^2}\]

Editorial & Symbolic Notes

Archival scan 0222.jpg from Fred W. Seelig’s notebook dossier on F.32 Markowski’s Theorem (Page 2 of 2). Companion OpenDocument worksheet(s): Circumcircle-Circumcenter, Incircle-Incenter. Classical Euclidean foundations: III.35, IV.5. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(3) — Circumcenter.