F.47 The UVW Coordinate Triangle — Folios

Archival manuscript scans and diplomatic transcriptions from seelig’s notebooks for F.47 (The UVW Coordinate Triangle) (SeeligNotebooks).

Folio 0295 (Page 1 of 11): The UVW Triangle — Coordinate & Intercept Setup (001)

[Folio 0295]

Folio 0295 (original scan 0295.jpg) from resources/seelig/47_u-v-w.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0295 (0295.jpg, Sheet 1 of 11):

Places \(A=(-U,0), B=(W,0), C=(0,V)\) around the altitude foot \((0,0)\) and writes the intercept equations of sides \(AC\) and \(BC\).

\[A = (-U, 0), \qquad B = (W, 0), \qquad C = (0, V), \qquad a^2 = V^2 + W^2,\; b^2 = U^2 + V^2,\; c = U + W\]

Editorial & Symbolic Notes

Archival scan 0295.jpg from Fred W. Seelig’s notebook dossier on F.47 The UVW Coordinate Triangle (Page 1 of 11). Companion OpenDocument worksheet(s): UVW Triangle, Ceva & Menelaus. Classical Euclidean foundations: I.47, VI.2. Kimberling Encyclopedia of Triangle Centers loci: X(2) — Centroid, X(4) — Orthocenter.

Folio 0296 (Page 2 of 11): 2x2 Matrix System for Cevian Trace D (002)

[Folio 0296]

Folio 0296 (original scan 0296.jpg) from resources/seelig/47_u-v-w.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0296 (0296.jpg, Sheet 2 of 11):

Intersects line \(AP\) through \(P(p,q)\) with side \(BC\) via Cramer’s rule to solve for \(D = (D_x, D_y)\).

\[F_x = \frac{p V}{V - q}, \qquad D = \left(\frac{W V (U + p)}{V(U + W) + q W},\; \frac{V q (U + W)}{V(U + W) + q W}\right)\]

Editorial & Symbolic Notes

Archival scan 0296.jpg from Fred W. Seelig’s notebook dossier on F.47 The UVW Coordinate Triangle (Page 2 of 11). Companion OpenDocument worksheet(s): UVW Triangle, Ceva & Menelaus. Classical Euclidean foundations: I.47, VI.2. Kimberling Encyclopedia of Triangle Centers loci: X(2) — Centroid, X(4) — Orthocenter.

Folio 0297 (Page 3 of 11): Heronian Juxtaposition Example (U, V, W) = (5, 12, 16) — Page 1 (003)

[Folio 0297]

Folio 0297 (original scan 0297.jpg) from resources/seelig/47_u-v-w.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0297 (0297.jpg, Sheet 3 of 11):

Combines \((5,12,13)\) and \((12,16,20)\) along shared altitude \(V=12\) to form the \((20,13,21)\) Heronian triangle (\(K=126\)).

\[A = (-U, 0), \qquad B = (W, 0), \qquad C = (0, V), \qquad a^2 = V^2 + W^2,\; b^2 = U^2 + V^2,\; c = U + W\]

Editorial & Symbolic Notes

Archival scan 0297.jpg from Fred W. Seelig’s notebook dossier on F.47 The UVW Coordinate Triangle (Page 3 of 11). Companion OpenDocument worksheet(s): UVW Triangle, Ceva & Menelaus. Classical Euclidean foundations: I.47, VI.2. Kimberling Encyclopedia of Triangle Centers loci: X(2) — Centroid, X(4) — Orthocenter.

Folio 0298 (Page 4 of 11): Heronian Juxtaposition Example — Centers & Radii (004)

[Folio 0298]

Folio 0298 (original scan 0298.jpg) from resources/seelig/47_u-v-w.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0298 (0298.jpg, Sheet 4 of 11):

Computes \(s=27, r = 14/3, R = 65/6\), and the coordinates of \(G, I, O, H\) in the \((5,12,16)\) UVW frame.

\[F_x = \frac{p V}{V - q}, \qquad D = \left(\frac{W V (U + p)}{V(U + W) + q W},\; \frac{V q (U + W)}{V(U + W) + q W}\right)\]

Editorial & Symbolic Notes

Archival scan 0298.jpg from Fred W. Seelig’s notebook dossier on F.47 The UVW Coordinate Triangle (Page 4 of 11). Companion OpenDocument worksheet(s): UVW Triangle, Ceva & Menelaus. Classical Euclidean foundations: I.47, VI.2. Kimberling Encyclopedia of Triangle Centers loci: X(2) — Centroid, X(4) — Orthocenter.

Folio 0299 (Page 5 of 11): 2x2 Matrix System for Cevian Trace E (005)

[Folio 0299]

Folio 0299 (original scan 0299.jpg) from resources/seelig/47_u-v-w.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0299 (0299.jpg, Sheet 5 of 11):

Intersects line \(BP\) with side \(AC\) to derive the closed-form coordinates of \(E = (E_x, E_y)\).

\[A = (-U, 0), \qquad B = (W, 0), \qquad C = (0, V), \qquad a^2 = V^2 + W^2,\; b^2 = U^2 + V^2,\; c = U + W\]

Editorial & Symbolic Notes

Archival scan 0299.jpg from Fred W. Seelig’s notebook dossier on F.47 The UVW Coordinate Triangle (Page 5 of 11). Companion OpenDocument worksheet(s): UVW Triangle, Ceva & Menelaus. Classical Euclidean foundations: I.47, VI.2. Kimberling Encyclopedia of Triangle Centers loci: X(2) — Centroid, X(4) — Orthocenter.

Folio 0300 (Page 6 of 11): Cevian Trace F on the Baseline & Ceva Ratios (006)

[Folio 0300]

Folio 0300 (original scan 0300.jpg) from resources/seelig/47_u-v-w.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0300 (0300.jpg, Sheet 6 of 11):

Finds \(F = (pV/(V-q), 0)\) on the \(x\)-axis and computes the signed Ceva ratios along all three sides.

\[F_x = \frac{p V}{V - q}, \qquad D = \left(\frac{W V (U + p)}{V(U + W) + q W},\; \frac{V q (U + W)}{V(U + W) + q W}\right)\]

Editorial & Symbolic Notes

Archival scan 0300.jpg from Fred W. Seelig’s notebook dossier on F.47 The UVW Coordinate Triangle (Page 6 of 11). Companion OpenDocument worksheet(s): UVW Triangle, Ceva & Menelaus. Classical Euclidean foundations: I.47, VI.2. Kimberling Encyclopedia of Triangle Centers loci: X(2) — Centroid, X(4) — Orthocenter.

Folio 0301 (Page 7 of 11): Worked Benchmark (U, V, W, p, q) = (8, 8, 16, 4, 4) — Page 1 (007)

[Folio 0301]

Folio 0301 (original scan 0301.jpg) from resources/seelig/47_u-v-w.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0301 (0301.jpg, Sheet 7 of 11):

Evaluates the matrix formulas at \(U=8, V=8, W=16, P=(4,4)\) to obtain \(D=(6.4, 4.8), E=(-2, 6), F=(8, 0)\).

\[A = (-U, 0), \qquad B = (W, 0), \qquad C = (0, V), \qquad a^2 = V^2 + W^2,\; b^2 = U^2 + V^2,\; c = U + W\]

Editorial & Symbolic Notes

Archival scan 0301.jpg from Fred W. Seelig’s notebook dossier on F.47 The UVW Coordinate Triangle (Page 7 of 11). Companion OpenDocument worksheet(s): UVW Triangle, Ceva & Menelaus. Classical Euclidean foundations: I.47, VI.2. Kimberling Encyclopedia of Triangle Centers loci: X(2) — Centroid, X(4) — Orthocenter.

Folio 0302 (Page 8 of 11): Worked Benchmark (8, 8, 16, 4, 4) — Ceva & Menelaus Check (008)

[Folio 0302]

Folio 0302 (original scan 0302.jpg) from resources/seelig/47_u-v-w.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0302 (0302.jpg, Sheet 8 of 11):

Verifies that the segment ratios determined by \(D, E, F\) multiply to \(1\).

\[F_x = \frac{p V}{V - q}, \qquad D = \left(\frac{W V (U + p)}{V(U + W) + q W},\; \frac{V q (U + W)}{V(U + W) + q W}\right)\]

Editorial & Symbolic Notes

Archival scan 0302.jpg from Fred W. Seelig’s notebook dossier on F.47 The UVW Coordinate Triangle (Page 8 of 11). Companion OpenDocument worksheet(s): UVW Triangle, Ceva & Menelaus. Classical Euclidean foundations: I.47, VI.2. Kimberling Encyclopedia of Triangle Centers loci: X(2) — Centroid, X(4) — Orthocenter.

Folio 0303 (Page 9 of 11): Barycentric Conversion in the UVW Frame (009)

[Folio 0303]

Folio 0303 (original scan 0303.jpg) from resources/seelig/47_u-v-w.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0303 (0303.jpg, Sheet 9 of 11):

Expresses the normalized barycentric coordinates \((\lambda_A, \lambda_B, \lambda_C)\) of \(P(p,q)\) in terms of \(U, V, W\).

\[A = (-U, 0), \qquad B = (W, 0), \qquad C = (0, V), \qquad a^2 = V^2 + W^2,\; b^2 = U^2 + V^2,\; c = U + W\]

Editorial & Symbolic Notes

Archival scan 0303.jpg from Fred W. Seelig’s notebook dossier on F.47 The UVW Coordinate Triangle (Page 9 of 11). Companion OpenDocument worksheet(s): UVW Triangle, Ceva & Menelaus. Classical Euclidean foundations: I.47, VI.2. Kimberling Encyclopedia of Triangle Centers loci: X(2) — Centroid, X(4) — Orthocenter.

Folio 0304 (Page 10 of 11): Inverse Mapping from (a, b, c) to (U, V, W) (010)

[Folio 0304]

Folio 0304 (original scan 0304.jpg) from resources/seelig/47_u-v-w.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0304 (0304.jpg, Sheet 10 of 11):

Uses Euclid II.12/II.13 to recover \(U = (b^2+c^2-a^2)/(2c), W = (a^2+c^2-b^2)/(2c)\), and \(V = 2K/c\).

\[F_x = \frac{p V}{V - q}, \qquad D = \left(\frac{W V (U + p)}{V(U + W) + q W},\; \frac{V q (U + W)}{V(U + W) + q W}\right)\]

Editorial & Symbolic Notes

Archival scan 0304.jpg from Fred W. Seelig’s notebook dossier on F.47 The UVW Coordinate Triangle (Page 10 of 11). Companion OpenDocument worksheet(s): UVW Triangle, Ceva & Menelaus. Classical Euclidean foundations: I.47, VI.2. Kimberling Encyclopedia of Triangle Centers loci: X(2) — Centroid, X(4) — Orthocenter.

Folio 0305 (Page 11 of 11): Spreadsheet Formula Summary for the UVW Worksheet (011)

[Folio 0305]

Folio 0305 (original scan 0305.jpg) from resources/seelig/47_u-v-w.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0305 (0305.jpg, Sheet 11 of 11):

Summarizes the cell formulas implemented in the UVW Triangle tab of Triangles and Circles.ods.

\[A = (-U, 0), \qquad B = (W, 0), \qquad C = (0, V), \qquad a^2 = V^2 + W^2,\; b^2 = U^2 + V^2,\; c = U + W\]

Editorial & Symbolic Notes

Archival scan 0305.jpg from Fred W. Seelig’s notebook dossier on F.47 The UVW Coordinate Triangle (Page 11 of 11). Companion OpenDocument worksheet(s): UVW Triangle, Ceva & Menelaus. Classical Euclidean foundations: I.47, VI.2. Kimberling Encyclopedia of Triangle Centers loci: X(2) — Centroid, X(4) — Orthocenter.