F.3 Biquadratic Forms in Heron’s Formula — Folios

Archival manuscript scans and diplomatic transcriptions from seelig’s notebooks for F.3 (Biquadratic Forms in Heron’s Formula) (SeeligNotebooks).

Folio 0011 (Page 1 of 9): Biquadratic Forms in Heron’s Formula

[Folio 0011]

Folio 0011 (original scan 0011.jpg) from resources/seelig/03_biquadratic-forms-in-herons-formula.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0011 (0011.jpg, Sheet 1 of 9):

Introduces \(K = \sqrt{s(s-a)(s-b)(s-c)}\) for a triangle inscribed in a circle of radius \(R\) with constant \(c\) and varying \(a\).

\[X = \frac{a}{2}, \quad W = \frac{b + c}{2}, \quad Y = \frac{c - b}{2} \implies K = \sqrt{(W^2 - X^2)(X^2 - Y^2)}\]

Editorial & Symbolic Notes

Archival scan 0011.jpg from Fred W. Seelig’s notebook dossier on F.3 Biquadratic Forms in Heron’s Formula (Page 1 of 9). Companion OpenDocument worksheet(s): General Reference, 5-6-7 A,B,C, Biquadratic Heron, Circumcircle Solver. Classical Euclidean foundations: I.47, II.12, II.13. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(3) — Circumcenter, X(4) — Orthocenter.

Folio 0012 (Page 2 of 9): Symmetric Substitutions W, X, Y, Z

[Folio 0012]

Folio 0012 (original scan 0012.jpg) from resources/seelig/03_biquadratic-forms-in-herons-formula.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0012 (0012.jpg, Sheet 2 of 9):

Defines \(X = a/2, W = (b+c)/2, Y = (-b+c)/2, Z = (b-c)/2\) to reduce Heron’s formula to \(K = \sqrt{(W^2 - X^2)(X^2 - Y^2)}\).

\[16 K^2 = 2 a^2 b^2 + 2 b^2 c^2 + 2 c^2 a^2 - a^4 - b^4 - c^4, \qquad {z_i^*}^2 = w_i^2 + w_i w_i' + {w_i'}^2\]

Editorial & Symbolic Notes

Archival scan 0012.jpg from Fred W. Seelig’s notebook dossier on F.3 Biquadratic Forms in Heron’s Formula (Page 2 of 9). Companion OpenDocument worksheet(s): General Reference, 5-6-7 A,B,C, Biquadratic Heron, Circumcircle Solver. Classical Euclidean foundations: I.47, II.12, II.13. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(3) — Circumcenter, X(4) — Orthocenter.

Folio 0013 (Page 3 of 9): Three Circumcircle Diagrams

[Folio 0013]

Folio 0013 (original scan 0013.jpg) from resources/seelig/03_biquadratic-forms-in-herons-formula.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0013 (0013.jpg, Sheet 3 of 9):

Compass-and-straightedge constructions showing perpendicular bisector distances \(h_A, h_B, h_C\), circumradii \(R\), and central angle \(2\beta\).

\[X = \frac{a}{2}, \quad W = \frac{b + c}{2}, \quad Y = \frac{c - b}{2} \implies K = \sqrt{(W^2 - X^2)(X^2 - Y^2)}\]

Editorial & Symbolic Notes

Archival scan 0013.jpg from Fred W. Seelig’s notebook dossier on F.3 Biquadratic Forms in Heron’s Formula (Page 3 of 9). Companion OpenDocument worksheet(s): General Reference, 5-6-7 A,B,C, Biquadratic Heron, Circumcircle Solver. Classical Euclidean foundations: I.47, II.12, II.13. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(3) — Circumcenter, X(4) — Orthocenter.

Folio 0014 (Page 4 of 9): Circumcircle Trigonometry & Law of Sines

[Folio 0014]

Folio 0014 (original scan 0014.jpg) from resources/seelig/03_biquadratic-forms-in-herons-formula.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0014 (0014.jpg, Sheet 4 of 9):

Records \(a/2 = R\sin\alpha, b/2 = R\sin\beta, c/2 = R\sin\gamma\) and \(R^2 = h_C^2 + (c/2)^2\).

\[16 K^2 = 2 a^2 b^2 + 2 b^2 c^2 + 2 c^2 a^2 - a^4 - b^4 - c^4, \qquad {z_i^*}^2 = w_i^2 + w_i w_i' + {w_i'}^2\]

Editorial & Symbolic Notes

Archival scan 0014.jpg from Fred W. Seelig’s notebook dossier on F.3 Biquadratic Forms in Heron’s Formula (Page 4 of 9). Companion OpenDocument worksheet(s): General Reference, 5-6-7 A,B,C, Biquadratic Heron, Circumcircle Solver. Classical Euclidean foundations: I.47, II.12, II.13. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(3) — Circumcenter, X(4) — Orthocenter.

Folio 0015 (Page 5 of 9): Finding Sidelength b(a; c, R)

[Folio 0015]

Folio 0015 (original scan 0015.jpg) from resources/seelig/03_biquadratic-forms-in-herons-formula.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0015 (0015.jpg, Sheet 5 of 9):

Derives \(b = 2R\sin(180^\circ - (\sin^{-1}\frac{a}{2R} + \sin^{-1}\frac{c}{2R}))\) for fixed chord \(c\) and circumradius \(R\).

\[X = \frac{a}{2}, \quad W = \frac{b + c}{2}, \quad Y = \frac{c - b}{2} \implies K = \sqrt{(W^2 - X^2)(X^2 - Y^2)}\]

Editorial & Symbolic Notes

Archival scan 0015.jpg from Fred W. Seelig’s notebook dossier on F.3 Biquadratic Forms in Heron’s Formula (Page 5 of 9). Companion OpenDocument worksheet(s): General Reference, 5-6-7 A,B,C, Biquadratic Heron, Circumcircle Solver. Classical Euclidean foundations: I.47, II.12, II.13. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(3) — Circumcenter, X(4) — Orthocenter.

Folio 0016 (Page 6 of 9): Heron’s Formula Grouped Around b/2

[Folio 0016]

Folio 0016 (original scan 0016.jpg) from resources/seelig/03_biquadratic-forms-in-herons-formula.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0016 (0016.jpg, Sheet 6 of 9):

Factors \(K = \sqrt{[(b/2)^2 - ((a-c)/2)^2][((a+c)/2)^2 - (b/2)^2]}\).

\[16 K^2 = 2 a^2 b^2 + 2 b^2 c^2 + 2 c^2 a^2 - a^4 - b^4 - c^4, \qquad {z_i^*}^2 = w_i^2 + w_i w_i' + {w_i'}^2\]

Editorial & Symbolic Notes

Archival scan 0016.jpg from Fred W. Seelig’s notebook dossier on F.3 Biquadratic Forms in Heron’s Formula (Page 6 of 9). Companion OpenDocument worksheet(s): General Reference, 5-6-7 A,B,C, Biquadratic Heron, Circumcircle Solver. Classical Euclidean foundations: I.47, II.12, II.13. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(3) — Circumcenter, X(4) — Orthocenter.

Folio 0017 (Page 7 of 9): Differentiation for Critical Value z*

[Folio 0017]

Folio 0017 (original scan 0017.jpg) from resources/seelig/03_biquadratic-forms-in-herons-formula.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0017 (0017.jpg, Sheet 7 of 9):

Differentiates \(f(z) = T_1 + T_2 + T_3 - \sqrt{3}E\) where \(T_i = \sqrt{((w_i/z)^2 - 1)(1 - (w_i'/z)^2)}\).

\[X = \frac{a}{2}, \quad W = \frac{b + c}{2}, \quad Y = \frac{c - b}{2} \implies K = \sqrt{(W^2 - X^2)(X^2 - Y^2)}\]

Editorial & Symbolic Notes

Archival scan 0017.jpg from Fred W. Seelig’s notebook dossier on F.3 Biquadratic Forms in Heron’s Formula (Page 7 of 9). Companion OpenDocument worksheet(s): General Reference, 5-6-7 A,B,C, Biquadratic Heron, Circumcircle Solver. Classical Euclidean foundations: I.47, II.12, II.13. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(3) — Circumcenter, X(4) — Orthocenter.

Folio 0018 (Page 8 of 9): Polynomial Division & Closed Form for z*

[Folio 0018]

Folio 0018 (original scan 0018.jpg) from resources/seelig/03_biquadratic-forms-in-herons-formula.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0018 (0018.jpg, Sheet 8 of 9):

Uses \((x^3 - y^3)/(x - y) = x^2 + xy + y^2\) to prove \({z_i^*}^2 = w_i^2 + w_i w_i' + {w_i'}^2\).

\[16 K^2 = 2 a^2 b^2 + 2 b^2 c^2 + 2 c^2 a^2 - a^4 - b^4 - c^4, \qquad {z_i^*}^2 = w_i^2 + w_i w_i' + {w_i'}^2\]

Editorial & Symbolic Notes

Archival scan 0018.jpg from Fred W. Seelig’s notebook dossier on F.3 Biquadratic Forms in Heron’s Formula (Page 8 of 9). Companion OpenDocument worksheet(s): General Reference, 5-6-7 A,B,C, Biquadratic Heron, Circumcircle Solver. Classical Euclidean foundations: I.47, II.12, II.13. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(3) — Circumcenter, X(4) — Orthocenter.

Folio 0019 (Page 9 of 9): Numerical Evaluation & Means of z_1*, z_2*, z_3*

[Folio 0019]

Folio 0019 (original scan 0019.jpg) from resources/seelig/03_biquadratic-forms-in-herons-formula.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0019 (0019.jpg, Sheet 9 of 9):

Computes \(z_1^* = \sqrt{91/4} \approx 4.77, z_2^* = \sqrt{57/4} \approx 3.775, z_3^* = \sqrt{21} \approx 4.583\) and compares their harmonic ($4.330$), geometric ($4.353$), and arithmetic ($4.376$) means.

\[X = \frac{a}{2}, \quad W = \frac{b + c}{2}, \quad Y = \frac{c - b}{2} \implies K = \sqrt{(W^2 - X^2)(X^2 - Y^2)}\]

Editorial & Symbolic Notes

Archival scan 0019.jpg from Fred W. Seelig’s notebook dossier on F.3 Biquadratic Forms in Heron’s Formula (Page 9 of 9). Companion OpenDocument worksheet(s): General Reference, 5-6-7 A,B,C, Biquadratic Heron, Circumcircle Solver. Classical Euclidean foundations: I.47, II.12, II.13. Kimberling Encyclopedia of Triangle Centers loci: X(1) — Incenter, X(3) — Circumcenter, X(4) — Orthocenter.