F.42 Altitude of a Pythagorean Triple Triangle — Folios

Archival manuscript scans and diplomatic transcriptions from seelig’s notebooks for F.42 (Altitude of a Pythagorean Triple Triangle) (SeeligNotebooks).

Folio 0267 (Page 1 of 4): Altitude of a Pythagorean Triple Triangle (001)

[Folio 0267]

Folio 0267 (original scan 0267.jpg) from resources/seelig/42_pythagorean-triples.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0267 (0267.jpg, Sheet 1 of 4):

Derives \(h_c = ab/c\), the Inverse Pythagorean Theorem \(h_c^{-2} = a^{-2} + b^{-2}\), and the hypotenuse segments \(u = b^2/c, y = a^2/c\).

\[h_c = \frac{a b}{c}, \qquad \frac{1}{h_c^2} = \frac{1}{a^2} + \frac{1}{b^2}, \qquad u = \frac{b^2}{c},\; y = \frac{a^2}{c},\; h_c^2 = u y\]

Editorial & Symbolic Notes

Archival scan 0267.jpg from Fred W. Seelig’s notebook dossier on F.42 Altitude of a Pythagorean Triple Triangle (Page 1 of 4). Companion OpenDocument worksheet(s): Pythagorean Triples, Right Triangle, Altitudes. Classical Euclidean foundations: I.47, VI.8. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(4) — Orthocenter, X(7) — Gergonne Point.

Folio 0268 (Page 2 of 4): Rational Evaluation on Primitive Triples (002)

[Folio 0268]

Folio 0268 (original scan 0268.jpg) from resources/seelig/42_pythagorean-triples.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0268 (0268.jpg, Sheet 2 of 4):

Evaluates \(h_c, u, y\) on \((3,4,5)\mapsto (12/5, 9/5, 16/5)\) and \((5,12,13)\mapsto (60/13, 25/13, 144/13)\).

\[|A T_C| = s - a \neq u = \frac{b^2}{c} \quad (a \neq b), \qquad K = (s - a)(s - b) = \frac{a b}{2}\]

Editorial & Symbolic Notes

Archival scan 0268.jpg from Fred W. Seelig’s notebook dossier on F.42 Altitude of a Pythagorean Triple Triangle (Page 2 of 4). Companion OpenDocument worksheet(s): Pythagorean Triples, Right Triangle, Altitudes. Classical Euclidean foundations: I.47, VI.8. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(4) — Orthocenter, X(7) — Gergonne Point.

Folio 0269 (Page 3 of 4): Self-Corrected Fallacy Box: CT_c vs. Altitude CH_c (003)

[Folio 0269]

Folio 0269 (original scan 0269.jpg) from resources/seelig/42_pythagorean-triples.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0269 (0269.jpg, Sheet 3 of 4):

Proves that the incircle touchpoint \(T_C\) (\(|AT_C| = s-a\)) is distinct from the altitude foot \(H_C\) (\(|AH_C| = b^2/c\)), boxing the warning ‘CT_c is not perp to BA’.

\[h_c = \frac{a b}{c}, \qquad \frac{1}{h_c^2} = \frac{1}{a^2} + \frac{1}{b^2}, \qquad u = \frac{b^2}{c},\; y = \frac{a^2}{c},\; h_c^2 = u y\]

Editorial & Symbolic Notes

Archival scan 0269.jpg from Fred W. Seelig’s notebook dossier on F.42 Altitude of a Pythagorean Triple Triangle (Page 3 of 4). Companion OpenDocument worksheet(s): Pythagorean Triples, Right Triangle, Altitudes. Classical Euclidean foundations: I.47, VI.8. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(4) — Orthocenter, X(7) — Gergonne Point.

Folio 0270 (Page 4 of 4): Orthocentric Projection Segments (004)

[Folio 0270]

Folio 0270 (original scan 0270.jpg) from resources/seelig/42_pythagorean-triples.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0270 (0270.jpg, Sheet 4 of 4):

Relates the right-triangle altitude segments to the general orthocentric system of an acute/obtuse triangle.

\[|A T_C| = s - a \neq u = \frac{b^2}{c} \quad (a \neq b), \qquad K = (s - a)(s - b) = \frac{a b}{2}\]

Editorial & Symbolic Notes

Archival scan 0270.jpg from Fred W. Seelig’s notebook dossier on F.42 Altitude of a Pythagorean Triple Triangle (Page 4 of 4). Companion OpenDocument worksheet(s): Pythagorean Triples, Right Triangle, Altitudes. Classical Euclidean foundations: I.47, VI.8. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(4) — Orthocenter, X(7) — Gergonne Point.