F.50 Altitudes, Mitchell’s Reciprocal Heron Formula, and the Symphonic Theorem — Folios

Archival manuscript scans and diplomatic transcriptions from seelig’s notebooks for F.50 (Altitudes, Mitchell’s Reciprocal Heron Formula, and the Symphonic Theorem) (SeeligNotebooks).

Folio 0318 (Page 1 of 4): Altitude of a Right Triangle — ‘The Sweet Theorem’ (altitude-right-001)

[Folio 0318]

Folio 0318 (original scan 0318.jpg) from resources/seelig/50_altitudes.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0318 (0318.jpg, Sheet 1 of 4):

Proves \(h = ab/c\) and the Inverse Pythagorean Theorem \(h^{-2} = a^{-2} + b^{-2}\).

\[H = \frac{h_A^{-1} + h_B^{-1} + h_C^{-1}}{2}, \qquad \frac{1}{K} = 4\sqrt{H(H - h_A^{-1})(H - h_B^{-1})(H - h_C^{-1})}\]

Editorial & Symbolic Notes

Archival scan 0318.jpg from Fred W. Seelig’s notebook dossier on F.50 Altitudes, Mitchell’s Reciprocal Heron Formula, and the Symphonic Theorem (Page 1 of 4). Companion OpenDocument worksheet(s): Altitudes, Orthocenter, Right Triangle. Classical Euclidean foundations: I.12, I.47, VI.8. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(3) — Circumcenter, X(4) — Orthocenter.

Folio 0319 (Page 2 of 4): Mitchell’s Reciprocal Heron Formula for Altitudes (altitudes-001)

[Folio 0319]

Folio 0319 (original scan 0319.jpg) from resources/seelig/50_altitudes.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0319 (0319.jpg, Sheet 2 of 4):

Cites Douglas W. Mitchell (2005) to express \(K^{-1} = 4\sqrt{H(H-h_A^{-1})(H-h_B^{-1})(H-h_C^{-1})}\) and reconstructs \((a,b,c) = (2K/h_A, 2K/h_B, 2K/h_C)\).

\[\frac{1}{h_A^2} + \frac{1}{h_B^2} = \frac{1}{h_C^2} \quad (\gamma = 90^\circ), \qquad |AH| + |BH| + |CH| = 2(r + R)\]

Editorial & Symbolic Notes

Archival scan 0319.jpg from Fred W. Seelig’s notebook dossier on F.50 Altitudes, Mitchell’s Reciprocal Heron Formula, and the Symphonic Theorem (Page 2 of 4). Companion OpenDocument worksheet(s): Altitudes, Orthocenter, Right Triangle. Classical Euclidean foundations: I.12, I.47, VI.8. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(3) — Circumcenter, X(4) — Orthocenter.

Folio 0320 (Page 3 of 4): The Symphonic Theorem & Altitude Ratios (altitudes-002)

[Folio 0320]

Folio 0320 (original scan 0320.jpg) from resources/seelig/50_altitudes.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0320 (0320.jpg, Sheet 3 of 4):

States the Symphonic Theorem \(h_A^{-2} + h_B^{-2} = h_C^{-2}\) for right triangles and relates \(h_A : h_B : h_C = a^{-1} : b^{-1} : c^{-1}\).

\[H = \frac{h_A^{-1} + h_B^{-1} + h_C^{-1}}{2}, \qquad \frac{1}{K} = 4\sqrt{H(H - h_A^{-1})(H - h_B^{-1})(H - h_C^{-1})}\]

Editorial & Symbolic Notes

Archival scan 0320.jpg from Fred W. Seelig’s notebook dossier on F.50 Altitudes, Mitchell’s Reciprocal Heron Formula, and the Symphonic Theorem (Page 3 of 4). Companion OpenDocument worksheet(s): Altitudes, Orthocenter, Right Triangle. Classical Euclidean foundations: I.12, I.47, VI.8. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(3) — Circumcenter, X(4) — Orthocenter.

Folio 0321 (Page 4 of 4): Six Orthocenter Segment Identities in Conway Notation (altitudes-003)

[Folio 0321]

Folio 0321 (original scan 0321.jpg) from resources/seelig/50_altitudes.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0321 (0321.jpg, Sheet 4 of 4):

Records \(t_A + t_B + t_C = 2(r+R)\) and five companion identities for the upper and lower orthocenter segments.

\[\frac{1}{h_A^2} + \frac{1}{h_B^2} = \frac{1}{h_C^2} \quad (\gamma = 90^\circ), \qquad |AH| + |BH| + |CH| = 2(r + R)\]

Editorial & Symbolic Notes

Archival scan 0321.jpg from Fred W. Seelig’s notebook dossier on F.50 Altitudes, Mitchell’s Reciprocal Heron Formula, and the Symphonic Theorem (Page 4 of 4). Companion OpenDocument worksheet(s): Altitudes, Orthocenter, Right Triangle. Classical Euclidean foundations: I.12, I.47, VI.8. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(3) — Circumcenter, X(4) — Orthocenter.