F.35 Obtuse and Acute Triangles — Folios

Archival manuscript scans and diplomatic transcriptions from seelig’s notebooks for F.35 (Obtuse and Acute Triangles) (SeeligNotebooks).

Folio 0225 (Page 1 of 7): Euclid’s Concept of Excess (II.12) & Deficit (II.13)

[Folio 0225]

Folio 0225 (original scan 0225.jpg) from resources/seelig/35_obtuse-and-acute.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0225 (0225.jpg, Sheet 1 of 7):

Formulates \(a^2 = b^2 + c^2 \pm 2cx\), solves for the projection \(x = \pm\frac{a^2-(b^2+c^2)}{2c}\), and recovers altitude \(h = \sqrt{b^2-x^2}\).

\[a^2 = b^2 + c^2 \pm 2 c x \implies x = \frac{|a^2 - (b^2 + c^2)|}{2 c}, \qquad h^2 = b^2 - x^2\]

Editorial & Symbolic Notes

Archival scan 0225.jpg from Fred W. Seelig’s notebook dossier on F.35 Obtuse and Acute Triangles (Page 1 of 7). Companion OpenDocument worksheet(s): General Reference, 5-6-7 A,B,C, Altitudes. Classical Euclidean foundations: II.12, II.13. Kimberling Encyclopedia of Triangle Centers loci: X(2) — Centroid, X(4) — Orthocenter.

Folio 0226 (Page 2 of 7): Obtuse (3, 5, 7) Triangle — Base 5 Projection

[Folio 0226]

Folio 0226 (original scan 0226.jpg) from resources/seelig/35_obtuse-and-acute.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0226 (0226.jpg, Sheet 2 of 7):

Evaluates \(a=7, b=3, c=5\) to find exterior projection \(x = 1.5\), altitude \(h_B = \frac{3\sqrt{3}}{2}\), and \(120^\circ\) obtuse angle.

\[|M_C H_C| = \left|\frac{c}{2} - x\right| = \frac{|a^2 - b^2|}{2 c}, \qquad m_c^2 = \frac{a^2 + b^2}{2} - \left(\frac{c}{2}\right)^2\]

Editorial & Symbolic Notes

Archival scan 0226.jpg from Fred W. Seelig’s notebook dossier on F.35 Obtuse and Acute Triangles (Page 2 of 7). Companion OpenDocument worksheet(s): General Reference, 5-6-7 A,B,C, Altitudes. Classical Euclidean foundations: II.12, II.13. Kimberling Encyclopedia of Triangle Centers loci: X(2) — Centroid, X(4) — Orthocenter.

Folio 0227 (Page 3 of 7): Obtuse (3, 5, 7) Triangle — Base 7 Projection

[Folio 0227]

Folio 0227 (original scan 0227.jpg) from resources/seelig/35_obtuse-and-acute.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0227 (0227.jpg, Sheet 3 of 7):

Projects onto the longest side \(c=7\) to find \(x = 65/14, h_c = \frac{15\sqrt{3}}{14}\), and area \(K = \frac{15\sqrt{3}}{4}\).

\[a^2 = b^2 + c^2 \pm 2 c x \implies x = \frac{|a^2 - (b^2 + c^2)|}{2 c}, \qquad h^2 = b^2 - x^2\]

Editorial & Symbolic Notes

Archival scan 0227.jpg from Fred W. Seelig’s notebook dossier on F.35 Obtuse and Acute Triangles (Page 3 of 7). Companion OpenDocument worksheet(s): General Reference, 5-6-7 A,B,C, Altitudes. Classical Euclidean foundations: II.12, II.13. Kimberling Encyclopedia of Triangle Centers loci: X(2) — Centroid, X(4) — Orthocenter.

Folio 0228 (Page 4 of 7): Acute (5, 6, 7) Triangle Deficit Analysis

[Folio 0228]

Folio 0228 (original scan 0228.jpg) from resources/seelig/35_obtuse-and-acute.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0228 (0228.jpg, Sheet 4 of 7):

Applies Euclid II.13 to \((5,6,7)\), finding interior projection \(x = 1\), altitude \(h = 2\sqrt{6}\), and segment ratio \(r/r' = 28/9\).

\[|M_C H_C| = \left|\frac{c}{2} - x\right| = \frac{|a^2 - b^2|}{2 c}, \qquad m_c^2 = \frac{a^2 + b^2}{2} - \left(\frac{c}{2}\right)^2\]

Editorial & Symbolic Notes

Archival scan 0228.jpg from Fred W. Seelig’s notebook dossier on F.35 Obtuse and Acute Triangles (Page 4 of 7). Companion OpenDocument worksheet(s): General Reference, 5-6-7 A,B,C, Altitudes. Classical Euclidean foundations: II.12, II.13. Kimberling Encyclopedia of Triangle Centers loci: X(2) — Centroid, X(4) — Orthocenter.

Folio 0229 (Page 5 of 7): Acute (7, 8, 9) Triangle — Altitude Foot Offsets (Page 1)

[Folio 0229]

Folio 0229 (original scan 0229.jpg) from resources/seelig/35_obtuse-and-acute.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0229 (0229.jpg, Sheet 5 of 7):

Computes the altitude feet \(h_A, h_B, h_C\) and midpoints \(m_A, m_B, m_C\) on the \((7,8,9)\) triangle.

\[a^2 = b^2 + c^2 \pm 2 c x \implies x = \frac{|a^2 - (b^2 + c^2)|}{2 c}, \qquad h^2 = b^2 - x^2\]

Editorial & Symbolic Notes

Archival scan 0229.jpg from Fred W. Seelig’s notebook dossier on F.35 Obtuse and Acute Triangles (Page 5 of 7). Companion OpenDocument worksheet(s): General Reference, 5-6-7 A,B,C, Altitudes. Classical Euclidean foundations: II.12, II.13. Kimberling Encyclopedia of Triangle Centers loci: X(2) — Centroid, X(4) — Orthocenter.

Folio 0230 (Page 6 of 7): Acute (7, 8, 9) Triangle — Midpoint-to-Foot Distances (Page 2)

[Folio 0230]

Folio 0230 (original scan 0230.jpg) from resources/seelig/35_obtuse-and-acute.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0230 (0230.jpg, Sheet 6 of 7):

Evaluates the exact rational offsets \(|m_A h_A| = 17/14, |m_B h_B| = 33/16\), and \(|m_C h_C| = 5/6\).

\[|M_C H_C| = \left|\frac{c}{2} - x\right| = \frac{|a^2 - b^2|}{2 c}, \qquad m_c^2 = \frac{a^2 + b^2}{2} - \left(\frac{c}{2}\right)^2\]

Editorial & Symbolic Notes

Archival scan 0230.jpg from Fred W. Seelig’s notebook dossier on F.35 Obtuse and Acute Triangles (Page 6 of 7). Companion OpenDocument worksheet(s): General Reference, 5-6-7 A,B,C, Altitudes. Classical Euclidean foundations: II.12, II.13. Kimberling Encyclopedia of Triangle Centers loci: X(2) — Centroid, X(4) — Orthocenter.

Folio 0231 (Page 7 of 7): Median Theorem from Excess & Deficit

[Folio 0231]

Folio 0231 (original scan 0231.jpg) from resources/seelig/35_obtuse-and-acute.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0231 (0231.jpg, Sheet 7 of 7):

Adds Euclid II.12 and II.13 on the two halves of a bisected base \(E\) to prove Apollonius’s median formula \(D^2 = \frac{U^2+V^2}{2} - (E/2)^2\).

\[a^2 = b^2 + c^2 \pm 2 c x \implies x = \frac{|a^2 - (b^2 + c^2)|}{2 c}, \qquad h^2 = b^2 - x^2\]

Editorial & Symbolic Notes

Archival scan 0231.jpg from Fred W. Seelig’s notebook dossier on F.35 Obtuse and Acute Triangles (Page 7 of 7). Companion OpenDocument worksheet(s): General Reference, 5-6-7 A,B,C, Altitudes. Classical Euclidean foundations: II.12, II.13. Kimberling Encyclopedia of Triangle Centers loci: X(2) — Centroid, X(4) — Orthocenter.