F.28 Incenter — Folios

Archival manuscript scans and diplomatic transcriptions from seelig’s notebooks for F.28 (Incenter) (SeeligNotebooks).

Folio 0202 (Page 1 of 5): Incenter of a Triangle Summary Sheet (FWS 6/1/2011)

[Folio 0202]

Folio 0202 (original scan 0202.jpg) from resources/seelig/28_incenter.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0202 (0202.jpg, Sheet 1 of 5):

Gives formulas for \(a, b, c, s, (X_O, Y_O), r = K/s\), and tangent segments \(s-a, s-b, s-c\).

\[\vec{I} = \frac{a \vec{A} + b \vec{B} + c \vec{C}}{a + b + c}, \qquad r = \frac{K}{s} = \sqrt{\frac{(s - a)(s - b)(s - c)}{s}}, \qquad \frac{1}{r} = \frac{1}{h_a} + \frac{1}{h_b} + \frac{1}{h_c}\]

Editorial & Symbolic Notes

Archival scan 0202.jpg from Fred W. Seelig’s notebook dossier on F.28 Incenter (Page 1 of 5). Companion OpenDocument worksheet(s): Incircle-Incenter, General Reference, 5-6-7 A,B,C. Classical Euclidean foundations: IV.4, VI.3. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(7) — Gergonne Point.

Folio 0203 (Page 2 of 5): Vector Equations for Contact Points T_A, T_B, T_C

[Folio 0203]

Folio 0203 (original scan 0203.jpg) from resources/seelig/28_incenter.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0203 (0203.jpg, Sheet 2 of 5):

Expresses \(\vec{T}_C = \vec{A} + \frac{x}{c}(\vec{B}-\vec{A})\) and line intersection formulas.

\[2 r R = \frac{a b c}{a + b + c}, \qquad d_{IO}^2 = R(R - 2 r) \implies R \ge 2 r\]

Editorial & Symbolic Notes

Archival scan 0203.jpg from Fred W. Seelig’s notebook dossier on F.28 Incenter (Page 2 of 5). Companion OpenDocument worksheet(s): Incircle-Incenter, General Reference, 5-6-7 A,B,C. Classical Euclidean foundations: IV.4, VI.3. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(7) — Gergonne Point.

Folio 0204 (Page 3 of 5): Inradius, Altitudes & Euler’s Formula (citing Altshiller-Court 2007)

[Folio 0204]

Folio 0204 (original scan 0204.jpg) from resources/seelig/28_incenter.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0204 (0204.jpg, Sheet 3 of 5):

Records \(1/r = 1/h_a + 1/h_b + 1/h_c\), \(2rR = abc/(a+b+c)\), and \(d^2 = R(R-2r)\).

\[\vec{I} = \frac{a \vec{A} + b \vec{B} + c \vec{C}}{a + b + c}, \qquad r = \frac{K}{s} = \sqrt{\frac{(s - a)(s - b)(s - c)}{s}}, \qquad \frac{1}{r} = \frac{1}{h_a} + \frac{1}{h_b} + \frac{1}{h_c}\]

Editorial & Symbolic Notes

Archival scan 0204.jpg from Fred W. Seelig’s notebook dossier on F.28 Incenter (Page 3 of 5). Companion OpenDocument worksheet(s): Incircle-Incenter, General Reference, 5-6-7 A,B,C. Classical Euclidean foundations: IV.4, VI.3. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(7) — Gergonne Point.

Folio 0205 (Page 4 of 5): Inscribed Circle in Unit Right Triangle

[Folio 0205]

Folio 0205 (original scan 0205.jpg) from resources/seelig/28_incenter.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0205 (0205.jpg, Sheet 4 of 5):

Graph-paper construction of the incircle and angle bisectors of the right triangle with vertices \((0,0), (0,1), (1,0)\).

\[2 r R = \frac{a b c}{a + b + c}, \qquad d_{IO}^2 = R(R - 2 r) \implies R \ge 2 r\]

Editorial & Symbolic Notes

Archival scan 0205.jpg from Fred W. Seelig’s notebook dossier on F.28 Incenter (Page 4 of 5). Companion OpenDocument worksheet(s): Incircle-Incenter, General Reference, 5-6-7 A,B,C. Classical Euclidean foundations: IV.4, VI.3. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(7) — Gergonne Point.

Folio 0206 (Page 5 of 5): Translating a Triangle so the Incenter is at the Origin (FWS Nov 8, 2011)

[Folio 0206]

Folio 0206 (original scan 0206.jpg) from resources/seelig/28_incenter.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0206 (0206.jpg, Sheet 5 of 5):

Derives shifted coordinates \(\vec{A} = -\vec{O}', \vec{B} = \vec{B}'-\vec{O}', \vec{C} = \vec{C}'-\vec{O}'\) with \(O'_x = bc(1+\cos\alpha)/(2s), O'_y = bc\sin\alpha/(2s)\).

\[\vec{I} = \frac{a \vec{A} + b \vec{B} + c \vec{C}}{a + b + c}, \qquad r = \frac{K}{s} = \sqrt{\frac{(s - a)(s - b)(s - c)}{s}}, \qquad \frac{1}{r} = \frac{1}{h_a} + \frac{1}{h_b} + \frac{1}{h_c}\]

Editorial & Symbolic Notes

Archival scan 0206.jpg from Fred W. Seelig’s notebook dossier on F.28 Incenter (Page 5 of 5). Companion OpenDocument worksheet(s): Incircle-Incenter, General Reference, 5-6-7 A,B,C. Classical Euclidean foundations: IV.4, VI.3. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(7) — Gergonne Point.