F.51 Apollonian Circles and the Problem of Apollonius — Folios

Archival manuscript scans and diplomatic transcriptions from seelig’s notebooks for F.51 (Apollonian Circles and the Problem of Apollonius) (SeeligNotebooks).

Folio 0322 (Page 1 of 9): Apollonian Circle Center M’ & Radius R (apollonian-001, FWS 2/7/2014)

[Folio 0322]

Folio 0322 (original scan 0322.jpg) from resources/seelig/51_apollonian-circles.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0322 (0322.jpg, Sheet 1 of 9):

Derives \(|BC| = d/(k+1), |BD| = d/(k-1), M' = d/(k^2-1)\), and \(R = dk/(k^2-1)\) for the locus \(|PA|/|PB| = k\).

\[\frac{|PA|}{|PB|} = k \implies M' = \frac{d}{k^2 - 1}, \qquad R = \frac{d k}{k^2 - 1}, \qquad R^2 - {M'}^2 = \frac{d^2}{k^2 - 1}\]

Editorial & Symbolic Notes

Archival scan 0322.jpg from Fred W. Seelig’s notebook dossier on F.51 Apollonian Circles and the Problem of Apollonius (Page 1 of 9). Companion OpenDocument worksheet(s): Apollonian Circles, Harmonic Conjugates, Coaxial Circles. Classical Euclidean foundations: III.36, VI.3.

Folio 0323 (Page 2 of 9): Cartesian Equation Completion of the Square (apollonian-002)

[Folio 0323]

Folio 0323 (original scan 0323.jpg) from resources/seelig/51_apollonian-circles.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0323 (0323.jpg, Sheet 2 of 9):

Expands \((x+d)^2 + y^2 = k^2(x^2+y^2)\) and completes the square to confirm center \((d/(k^2-1), 0)\) and radius \(dk/(k^2-1)\).

\[|A T'|^2 = |AC|\cdot|AD| = |AB|\cdot|A M'| = d(d + M') = \frac{d^2 k^2}{k^2 - 1}\]

Editorial & Symbolic Notes

Archival scan 0323.jpg from Fred W. Seelig’s notebook dossier on F.51 Apollonian Circles and the Problem of Apollonius (Page 2 of 9). Companion OpenDocument worksheet(s): Apollonian Circles, Harmonic Conjugates, Coaxial Circles. Classical Euclidean foundations: III.36, VI.3.

Folio 0324 (Page 3 of 9): Angle Bisector Proof via Euclid VI.3 (apollonian-003)

[Folio 0324]

Folio 0324 (original scan 0324.jpg) from resources/seelig/51_apollonian-circles.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0324 (0324.jpg, Sheet 3 of 9):

Shows that the internal and external bisectors of \(\angle APB\) meet \(AB\) at fixed harmonic points \(C, D\) with \(\angle CPD = 90^\circ\).

\[\frac{|PA|}{|PB|} = k \implies M' = \frac{d}{k^2 - 1}, \qquad R = \frac{d k}{k^2 - 1}, \qquad R^2 - {M'}^2 = \frac{d^2}{k^2 - 1}\]

Editorial & Symbolic Notes

Archival scan 0324.jpg from Fred W. Seelig’s notebook dossier on F.51 Apollonian Circles and the Problem of Apollonius (Page 3 of 9). Companion OpenDocument worksheet(s): Apollonian Circles, Harmonic Conjugates, Coaxial Circles. Classical Euclidean foundations: III.36, VI.3.

Folio 0325 (Page 4 of 9): Power of Point A & Tangent Length |AT’| (apollonian-004)

[Folio 0325]

Folio 0325 (original scan 0325.jpg) from resources/seelig/51_apollonian-circles.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0325 (0325.jpg, Sheet 4 of 9):

Proves \(|AT'|^2 = |AC|\cdot|AD| = |AB|\cdot|AM'|\) for the tangent \(AT'\) from \(A\) to the Apollonian circle.

\[|A T'|^2 = |AC|\cdot|AD| = |AB|\cdot|A M'| = d(d + M') = \frac{d^2 k^2}{k^2 - 1}\]

Editorial & Symbolic Notes

Archival scan 0325.jpg from Fred W. Seelig’s notebook dossier on F.51 Apollonian Circles and the Problem of Apollonius (Page 4 of 9). Companion OpenDocument worksheet(s): Apollonian Circles, Harmonic Conjugates, Coaxial Circles. Classical Euclidean foundations: III.36, VI.3.

Folio 0326 (Page 5 of 9): Orthogonal Circle of Inversion Centered at A or B (apollonian-005)

[Folio 0326]

Folio 0326 (original scan 0326.jpg) from resources/seelig/51_apollonian-circles.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0326 (0326.jpg, Sheet 5 of 9):

Shows that the circle centered at \(A\) of radius \(|AT'| = dk/\sqrt{k^2-1}\) is orthogonal to the Apollonian circle.

\[\frac{|PA|}{|PB|} = k \implies M' = \frac{d}{k^2 - 1}, \qquad R = \frac{d k}{k^2 - 1}, \qquad R^2 - {M'}^2 = \frac{d^2}{k^2 - 1}\]

Editorial & Symbolic Notes

Archival scan 0326.jpg from Fred W. Seelig’s notebook dossier on F.51 Apollonian Circles and the Problem of Apollonius (Page 5 of 9). Companion OpenDocument worksheet(s): Apollonian Circles, Harmonic Conjugates, Coaxial Circles. Classical Euclidean foundations: III.36, VI.3.

Folio 0327 (Page 6 of 9): Arithmetic, Geometric & Harmonic Means on Segment AD (apollonian-006)

[Folio 0327]

Folio 0327 (original scan 0327.jpg) from resources/seelig/51_apollonian-circles.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0327 (0327.jpg, Sheet 6 of 9):

Identifies \(|AB|, |AC|, |AD|, |AM'|\) with the classical Pythagorean means.

\[|A T'|^2 = |AC|\cdot|AD| = |AB|\cdot|A M'| = d(d + M') = \frac{d^2 k^2}{k^2 - 1}\]

Editorial & Symbolic Notes

Archival scan 0327.jpg from Fred W. Seelig’s notebook dossier on F.51 Apollonian Circles and the Problem of Apollonius (Page 6 of 9). Companion OpenDocument worksheet(s): Apollonian Circles, Harmonic Conjugates, Coaxial Circles. Classical Euclidean foundations: III.36, VI.3.

Folio 0328 (Page 7 of 9): Integer Mean Search: m = 36, n = 324 (apollonian-007)

[Folio 0328]

Folio 0328 (original scan 0328.jpg) from resources/seelig/51_apollonian-circles.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0328 (0328.jpg, Sheet 7 of 9):

Solves for integer pairs \((m,n)\) having integer arithmetic mean \(A = 180\) and geometric mean \(G = 108\).

\[\frac{|PA|}{|PB|} = k \implies M' = \frac{d}{k^2 - 1}, \qquad R = \frac{d k}{k^2 - 1}, \qquad R^2 - {M'}^2 = \frac{d^2}{k^2 - 1}\]

Editorial & Symbolic Notes

Archival scan 0328.jpg from Fred W. Seelig’s notebook dossier on F.51 Apollonian Circles and the Problem of Apollonius (Page 7 of 9). Companion OpenDocument worksheet(s): Apollonian Circles, Harmonic Conjugates, Coaxial Circles. Classical Euclidean foundations: III.36, VI.3.

Folio 0329 (Page 8 of 9): Coaxial Family of Apollonian Circles for Varying k (apollonian-008)

[Folio 0329]

Folio 0329 (original scan 0329.jpg) from resources/seelig/51_apollonian-circles.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0329 (0329.jpg, Sheet 8 of 9):

Plots the nested Apollonian circles around limiting points \(A\) and \(B\) with the perpendicular bisector as the \(k=1\) radical axis.

\[|A T'|^2 = |AC|\cdot|AD| = |AB|\cdot|A M'| = d(d + M') = \frac{d^2 k^2}{k^2 - 1}\]

Editorial & Symbolic Notes

Archival scan 0329.jpg from Fred W. Seelig’s notebook dossier on F.51 Apollonian Circles and the Problem of Apollonius (Page 8 of 9). Companion OpenDocument worksheet(s): Apollonian Circles, Harmonic Conjugates, Coaxial Circles. Classical Euclidean foundations: III.36, VI.3.

Folio 0330 (Page 9 of 9): Problem of Apollonius — Dörrie’s 8-Circle I/X Tangency Table (apollonius-2-001)

[Folio 0330]

Folio 0330 (original scan 0330.jpg) from resources/seelig/51_apollonian-circles.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0330 (0330.jpg, Sheet 9 of 9):

Cites Heinrich Dörrie (#32, pp. 154–160) and tabulates all 8 solution circles by their internal (I) and external (X) tangency states to three given circles.

\[\frac{|PA|}{|PB|} = k \implies M' = \frac{d}{k^2 - 1}, \qquad R = \frac{d k}{k^2 - 1}, \qquad R^2 - {M'}^2 = \frac{d^2}{k^2 - 1}\]

Editorial & Symbolic Notes

Archival scan 0330.jpg from Fred W. Seelig’s notebook dossier on F.51 Apollonian Circles and the Problem of Apollonius (Page 9 of 9). Companion OpenDocument worksheet(s): Apollonian Circles, Harmonic Conjugates, Coaxial Circles. Classical Euclidean foundations: III.36, VI.3.