Descartes' theorem (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Descartes' theorem" in English language version.

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americanscientist.org

Mackenzie, Dana (March–April 2023), "The princess and the philosopher", American Scientist, vol. 111, no. 2, pp. 80–84, ProQuest 2779946948

ams.org

mathscinet.ams.org

Hogendijk, Jan P. (1986), "Arabic traces of lost works of Apollonius", Archive for History of Exact Sciences, 35 (3): 187–253, doi:10.1007/BF00357307, JSTOR 41133783, MR 0851067, S2CID 121613986
Barnes, John (2012), "Soddy's hexlet", Gems of Geometry (2nd ed.), Heidelberg: Springer, pp. 173–177, doi:10.1007/978-3-642-30964-9, ISBN 978-3-642-30963-2, MR 2963305
Lagarias, Jeffrey C.; Mallows, Colin L.; Wilks, Allan R. (2002), "Beyond the Descartes circle theorem", The American Mathematical Monthly, 109 (4): 338–361, arXiv:math/0101066, doi:10.2307/2695498, JSTOR 2695498, MR 1903421
Levrie, Paul (2019), "A straightforward proof of Descartes's circle theorem", The Mathematical Intelligencer, 41 (3): 24–27, doi:10.1007/s00283-019-09883-x, hdl:10067/1621880151162165141, MR 3995314, S2CID 253818666
Pedoe, Daniel (1967), "On a theorem in geometry", The American Mathematical Monthly, 74 (6): 627–640, doi:10.2307/2314247, JSTOR 2314247, MR 0215169
Bradford, Alden (2023), "An even more straightforward proof of Descartes's circle theorem", The Mathematical Intelligencer, 45 (3): 263–265, arXiv:2211.05539, doi:10.1007/s00283-022-10234-6, MR 4645170
Wilker, J. B. (1969), "Four proofs of a generalization of the Descartes circle theorem", The American Mathematical Monthly, 76 (3): 278–282, doi:10.2307/2316373, JSTOR 2316373, MR 0246207
Northshield, Sam (2014), "Complex Descartes circle theorem", The American Mathematical Monthly, 121 (10): 927–931, doi:10.4169/amer.math.monthly.121.10.927, hdl:1951/69912, JSTOR 10.4169/amer.math.monthly.121.10.927, MR 3295667, S2CID 16335704
Tupan, Alexandru (2022), "On the complex Descartes circle theorem", The American Mathematical Monthly, 129 (9): 876–879, doi:10.1080/00029890.2022.2104084, MR 4499753, S2CID 251417228
Dergiades, Nikolaos (2007), "The Soddy circles" (PDF), Forum Geometricorum, 7: 191–197, MR 2373402
Graham, Ronald L.; Lagarias, Jeffrey C.; Mallows, Colin L.; Wilks, Allan R.; Yan, Catherine H. (2003), "Apollonian circle packings: number theory", Journal of Number Theory, 100 (1): 1–45, arXiv:math/0009113, doi:10.1016/S0022-314X(03)00015-5, MR 1971245, S2CID 16607718
McGonagle, Annmarie; Northshield, Sam (2014), "A new parameterization of Ford circles", Pi Mu Epsilon Journal, 13 (10): 637–643, JSTOR 24345283, MR 3235834
Coxeter, H. S. M. (1968), "Loxodromic sequences of tangent spheres", Aequationes Mathematicae, 1 (1–2): 104–121, doi:10.1007/BF01817563, MR 0235456, S2CID 119897862
This concept of distance was called the "pseudo-chordal distance" for the complex unit disk as a model for the hyperbolic plane by Carathéodory, Constantin (1954) [1950], "§§1.3.86–88 Chordal and Pseudo-chordal Distance", Theory of Functions of a Complex Variable, vol. I, translated by Steinhardt, Fritz, Chelsea, pp. 81–86, MR 0060009

archive.org

Descartes, René (1901), Adam, Charles; Tannery, Paul (eds.), Oeuvres de Descartes (in French), vol. 4: Correspondance Juillet 1643 – Avril 1647, Paris: Léopold Cerf, "325. Descartes a Elisabeth", pp. 37–42; "328. Descartes a Elisabeth", pp. 45–50
Bos, Erik-Jan (2010), "Princess Elizabeth of Bohemia and Descartes' letters (1650–1665)", Historia Mathematica, 37 (3): 485–502, doi:10.1016/j.hm.2009.11.004
Steiner, Jakob (January 1826), "Fortsetzung der geometrischen Betrachtungen (Heft 2, S. 161)", Journal für die reine und angewandte Mathematik, 1826 (1), pp. 252–288, fig. 2–25 taf. III, doi:10.1515/crll.1826.1.252, S2CID 121590578
Weiss, Asia (1981), "On Coxeter's Loxodromic Sequences of Tangent Spheres", in Davis, Chandler; Grünbaum, Branko; Sherk, F.A. (eds.), The Geometric Vein: The Coxeter Festschrift, Springer, pp. 241–250, doi:10.1007/978-1-4612-5648-9_16, ISBN 978-1-4612-5650-2
Coolidge, Julian Lowell (1916), "X. The Oriented Circle", A Treatise on the Circle and the Sphere, Clarendon, pp. 351–407, also see p. 109, p. 408
Rigby, J. F. (1981), "The geometry of cycles, and generalized Laguerre inversion", in Davis, Chandler; Grünbaum, Branko; Sherk, F.A. (eds.), The Geometric Vein: The Coxeter Festschrift, Springer, pp. 355–378, doi:10.1007/978-1-4612-5648-9_26, ISBN 978-1-4612-5650-2
This concept of distance was called the "pseudo-chordal distance" for the complex unit disk as a model for the hyperbolic plane by Carathéodory, Constantin (1954) [1950], "§§1.3.86–88 Chordal and Pseudo-chordal Distance", Theory of Functions of a Complex Variable, vol. I, translated by Steinhardt, Fritz, Chelsea, pp. 81–86, MR 0060009

arxiv.org

Lagarias, Jeffrey C.; Mallows, Colin L.; Wilks, Allan R. (2002), "Beyond the Descartes circle theorem", The American Mathematical Monthly, 109 (4): 338–361, arXiv:math/0101066, doi:10.2307/2695498, JSTOR 2695498, MR 1903421
Bradford, Alden (2023), "An even more straightforward proof of Descartes's circle theorem", The Mathematical Intelligencer, 45 (3): 263–265, arXiv:2211.05539, doi:10.1007/s00283-022-10234-6, MR 4645170
Graham, Ronald L.; Lagarias, Jeffrey C.; Mallows, Colin L.; Wilks, Allan R.; Yan, Catherine H. (2003), "Apollonian circle packings: number theory", Journal of Number Theory, 100 (1): 1–45, arXiv:math/0009113, doi:10.1016/S0022-314X(03)00015-5, MR 1971245, S2CID 16607718
Garcia, Ronaldo; Reznik, Dan; Moses, Peter; Gheorghe, Liliana (2022), "Triads of conics associated with a triangle", KoG (26), Croatian Society for Geometry and Graphics: 16–32, arXiv:2112.15232, doi:10.31896/k.26.2, S2CID 245634505
Kocik, Jerzy (2007), A theorem on circle configurations, arXiv:0706.0372
Kocik, Jerzy (2010), "Golden window" (PDF), Mathematics Magazine, 83 (5): 384–390, JSTOR 10.4169/002557010x529815, doi:10.4169/002557010x529815

Kocik, Jerzy (2019), Proof of Descartes circle formula and its generalization clarified, arXiv:1910.09174
Eriksson, Nicholas; Lagarias, Jeffrey C. (2007), "Apollonian Circle Packings: Number Theory II. Spherical and Hyperbolic Packings", The Ramanujan Journal, 14 (3): 437–469, arXiv:math/0403296, doi:10.1007/s11139-007-9052-6, S2CID 14024662

bnf.fr

gallica.bnf.fr

Lemoine, Émile (1891), "Sur les triangles orthologiques et sur divers sujets de la géométrie du triangle" [On orthologic triangles and on various subjects of triangle geometry], Compte rendu de la 19me session de l'association française pour l'avancement des sciences, pt. 2, Congrès de Limoges 1890 (in French), Paris: Secrétariat de l'association, pp. 111–146, especially §4 "Sur les intersections deux a deux des coniques qui ont pour foyers-deux sommets d'un triangle et passent par le troisième" [On the intersections in pairs of the conics which have as foci two vertices of a triangle and pass through the third], pp. 128–144

bridgesmathart.org

archive.bridgesmathart.org

Hidetoshi, Fukagawa; Kazunori, Horibe (2014), "Sangaku – Japanese Mathematics and Art in the 18th, 19th and 20th Centuries", in Greenfield, Gary; Hart, George; Sarhangi, Reza (eds.), Bridges Seoul Conference Proceedings, Tessellations Publishing, pp. 111–118

davidhestenes.net

A definition of stereographic distance can be found in Li, Hongbo; Hestenes, David; Rockwood, Alyn (2001), "Spherical conformal geometry with geometric algebra" (PDF), Geometric Computing with Clifford Algebras, Springer, pp. 61–75, CiteSeerX 10.1.1.412.4949, doi:10.1007/978-3-662-04621-0_3, ISBN 978-3-642-07442-4

doi.org

Soddy, F. (June 1936), "The Kiss Precise", Nature, 137 (3477): 1021, Bibcode:1936Natur.137.1021S, doi:10.1038/1371021a0, S2CID 6012051
Hogendijk, Jan P. (1986), "Arabic traces of lost works of Apollonius", Archive for History of Exact Sciences, 35 (3): 187–253, doi:10.1007/BF00357307, JSTOR 41133783, MR 0851067, S2CID 121613986
Court, Nathan Altshiller (October 1961), "The problem of Apollonius", The Mathematics Teacher, 54 (6): 444–452, doi:10.5951/MT.54.6.0444, JSTOR 27956431
Descartes, René (1901), Adam, Charles; Tannery, Paul (eds.), Oeuvres de Descartes (in French), vol. 4: Correspondance Juillet 1643 – Avril 1647, Paris: Léopold Cerf, "325. Descartes a Elisabeth", pp. 37–42; "328. Descartes a Elisabeth", pp. 45–50
Bos, Erik-Jan (2010), "Princess Elizabeth of Bohemia and Descartes' letters (1650–1665)", Historia Mathematica, 37 (3): 485–502, doi:10.1016/j.hm.2009.11.004
Coxeter, H. S. M. (January 1968), "The problem of Apollonius", The American Mathematical Monthly, 75 (1): 5–15, doi:10.1080/00029890.1968.11970941, JSTOR 2315097
Michiwaki, Yoshimasa (2008), "Geometry in Japanese mathematics", in Selin, Helaine (ed.), Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, Springer Netherlands, pp. 1018–1019, doi:10.1007/978-1-4020-4425-0_9133, ISBN 978-1-4020-4559-2
Rothman, Tony; Fugakawa, Hidetoshi (May 1998), "Japanese temple geometry", Scientific American, 278 (5): 84–91, Bibcode:1998SciAm.278e..84R, doi:10.1038/scientificamerican0598-84, JSTOR 26057787; see top illustration, p. 86. Another tablet from 1822 (center, p. 88) concerns Soddy's hexlet, a configuration of three-dimensional tangent spheres.
Steiner, Jakob (January 1826), "Fortsetzung der geometrischen Betrachtungen (Heft 2, S. 161)", Journal für die reine und angewandte Mathematik, 1826 (1), pp. 252–288, fig. 2–25 taf. III, doi:10.1515/crll.1826.1.252, S2CID 121590578
Soddy, Frederick (December 1936), "The hexlet", Nature, 138 (3501): 958, Bibcode:1936Natur.138..958S, doi:10.1038/138958a0, S2CID 28170211
Barnes, John (2012), "Soddy's hexlet", Gems of Geometry (2nd ed.), Heidelberg: Springer, pp. 173–177, doi:10.1007/978-3-642-30964-9, ISBN 978-3-642-30963-2, MR 2963305
Gardner, Martin (May 1968), "Mathematical Games: Circles and spheres, and how they kiss and pack", Scientific American, 218 (5): 130–139, doi:10.1038/scientificamerican0568-130, JSTOR 24926234
"The Kiss Precise", Nature, 139 (3506): 62, January 1937, Bibcode:1937Natur.139Q..62., doi:10.1038/139062a0
Lagarias, Jeffrey C.; Mallows, Colin L.; Wilks, Allan R. (2002), "Beyond the Descartes circle theorem", The American Mathematical Monthly, 109 (4): 338–361, arXiv:math/0101066, doi:10.2307/2695498, JSTOR 2695498, MR 1903421
Levrie, Paul (2019), "A straightforward proof of Descartes's circle theorem", The Mathematical Intelligencer, 41 (3): 24–27, doi:10.1007/s00283-019-09883-x, hdl:10067/1621880151162165141, MR 3995314, S2CID 253818666
Pedoe, Daniel (1967), "On a theorem in geometry", The American Mathematical Monthly, 74 (6): 627–640, doi:10.2307/2314247, JSTOR 2314247, MR 0215169
Bradford, Alden (2023), "An even more straightforward proof of Descartes's circle theorem", The Mathematical Intelligencer, 45 (3): 263–265, arXiv:2211.05539, doi:10.1007/s00283-022-10234-6, MR 4645170
Wilker, J. B. (1969), "Four proofs of a generalization of the Descartes circle theorem", The American Mathematical Monthly, 76 (3): 278–282, doi:10.2307/2316373, JSTOR 2316373, MR 0246207
Glaeser, Georg; Stachel, Hellmuth; Odehnal, Boris (2016), "The parabolic pencil – a common line element", The Universe of Conics, Springer, p. 327, doi:10.1007/978-3-662-45450-3, ISBN 978-3-662-45449-7
Northshield, Sam (2014), "Complex Descartes circle theorem", The American Mathematical Monthly, 121 (10): 927–931, doi:10.4169/amer.math.monthly.121.10.927, hdl:1951/69912, JSTOR 10.4169/amer.math.monthly.121.10.927, MR 3295667, S2CID 16335704
Tupan, Alexandru (2022), "On the complex Descartes circle theorem", The American Mathematical Monthly, 129 (9): 876–879, doi:10.1080/00029890.2022.2104084, MR 4499753, S2CID 251417228
This is a special case of a formula for the radii of circles in a Steiner chain with concentric inner and outer circles, given by Sheydvasser, Arseniy (2023), "3.1 Steiner's porism and 3.6 Steiner's porism revisited", Linear Fractional Transformations, Undergraduate Texts in Mathematics, Springer International Publishing, pp. 75–81, 99–101, doi:10.1007/978-3-031-25002-6, ISBN 978-3-031-25001-9, S2CID 258177153
Graham, Ronald L.; Lagarias, Jeffrey C.; Mallows, Colin L.; Wilks, Allan R.; Yan, Catherine H. (2003), "Apollonian circle packings: number theory", Journal of Number Theory, 100 (1): 1–45, arXiv:math/0009113, doi:10.1016/S0022-314X(03)00015-5, MR 1971245, S2CID 16607718
Bradley, Christopher J. (March 2003), "Heron triangles and touching circles", The Mathematical Gazette, 87 (508): 36–41, doi:10.1017/s0025557200172080, JSTOR 3620562, S2CID 125024700
Coxeter, H. S. M. (1968), "Loxodromic sequences of tangent spheres", Aequationes Mathematicae, 1 (1–2): 104–121, doi:10.1007/BF01817563, MR 0235456, S2CID 119897862
Weiss, Asia (1981), "On Coxeter's Loxodromic Sequences of Tangent Spheres", in Davis, Chandler; Grünbaum, Branko; Sherk, F.A. (eds.), The Geometric Vein: The Coxeter Festschrift, Springer, pp. 241–250, doi:10.1007/978-1-4612-5648-9_16, ISBN 978-1-4612-5650-2
Veldkamp, G. R. (1985), "The Isoperimetric Point and the Point(s) of Equal Detour in a Triangle", The American Mathematical Monthly, 92 (8): 546–558, doi:10.1080/00029890.1985.11971677, JSTOR 2323159
Garcia, Ronaldo; Reznik, Dan; Moses, Peter; Gheorghe, Liliana (2022), "Triads of conics associated with a triangle", KoG (26), Croatian Society for Geometry and Graphics: 16–32, arXiv:2112.15232, doi:10.31896/k.26.2, S2CID 245634505
Mauldon, J. G. (1962), "Sets of equally inclined spheres", Canadian Journal of Mathematics, 14: 509–516, doi:10.4153/CJM-1962-042-6
Rigby, J. F. (1981), "The geometry of cycles, and generalized Laguerre inversion", in Davis, Chandler; Grünbaum, Branko; Sherk, F.A. (eds.), The Geometric Vein: The Coxeter Festschrift, Springer, pp. 355–378, doi:10.1007/978-1-4612-5648-9_26, ISBN 978-1-4612-5650-2
A definition of stereographic distance can be found in Li, Hongbo; Hestenes, David; Rockwood, Alyn (2001), "Spherical conformal geometry with geometric algebra" (PDF), Geometric Computing with Clifford Algebras, Springer, pp. 61–75, CiteSeerX 10.1.1.412.4949, doi:10.1007/978-3-662-04621-0_3, ISBN 978-3-642-07442-4
Eriksson, Nicholas; Lagarias, Jeffrey C. (2007), "Apollonian Circle Packings: Number Theory II. Spherical and Hyperbolic Packings", The Ramanujan Journal, 14 (3): 437–469, arXiv:math/0403296, doi:10.1007/s11139-007-9052-6, S2CID 14024662

fau.edu

forumgeom.fau.edu

Dergiades, Nikolaos (2007), "The Soddy circles" (PDF), Forum Geometricorum, 7: 191–197, MR 2373402
Jackson, Frank M. (2013), "Soddyian Triangles" (PDF), Forum Geometricorum, 13: 1–6

handle.net

hdl.handle.net

Levrie, Paul (2019), "A straightforward proof of Descartes's circle theorem", The Mathematical Intelligencer, 41 (3): 24–27, doi:10.1007/s00283-019-09883-x, hdl:10067/1621880151162165141, MR 3995314, S2CID 253818666
Northshield, Sam (2014), "Complex Descartes circle theorem", The American Mathematical Monthly, 121 (10): 927–931, doi:10.4169/amer.math.monthly.121.10.927, hdl:1951/69912, JSTOR 10.4169/amer.math.monthly.121.10.927, MR 3295667, S2CID 16335704

harvard.edu

ui.adsabs.harvard.edu

Soddy, F. (June 1936), "The Kiss Precise", Nature, 137 (3477): 1021, Bibcode:1936Natur.137.1021S, doi:10.1038/1371021a0, S2CID 6012051
Rothman, Tony; Fugakawa, Hidetoshi (May 1998), "Japanese temple geometry", Scientific American, 278 (5): 84–91, Bibcode:1998SciAm.278e..84R, doi:10.1038/scientificamerican0598-84, JSTOR 26057787; see top illustration, p. 86. Another tablet from 1822 (center, p. 88) concerns Soddy's hexlet, a configuration of three-dimensional tangent spheres.
Soddy, Frederick (December 1936), "The hexlet", Nature, 138 (3501): 958, Bibcode:1936Natur.138..958S, doi:10.1038/138958a0, S2CID 28170211
"The Kiss Precise", Nature, 139 (3506): 62, January 1937, Bibcode:1937Natur.139Q..62., doi:10.1038/139062a0

hathitrust.org

babel.hathitrust.org

Beecroft, Philip (1842), "Properties of circles in mutual contact", The Lady's and Gentleman's Diary (139): 91–96

jstor.org

Hogendijk, Jan P. (1986), "Arabic traces of lost works of Apollonius", Archive for History of Exact Sciences, 35 (3): 187–253, doi:10.1007/BF00357307, JSTOR 41133783, MR 0851067, S2CID 121613986
Court, Nathan Altshiller (October 1961), "The problem of Apollonius", The Mathematics Teacher, 54 (6): 444–452, doi:10.5951/MT.54.6.0444, JSTOR 27956431
Coxeter, H. S. M. (January 1968), "The problem of Apollonius", The American Mathematical Monthly, 75 (1): 5–15, doi:10.1080/00029890.1968.11970941, JSTOR 2315097
Rothman, Tony; Fugakawa, Hidetoshi (May 1998), "Japanese temple geometry", Scientific American, 278 (5): 84–91, Bibcode:1998SciAm.278e..84R, doi:10.1038/scientificamerican0598-84, JSTOR 26057787; see top illustration, p. 86. Another tablet from 1822 (center, p. 88) concerns Soddy's hexlet, a configuration of three-dimensional tangent spheres.
Gardner, Martin (May 1968), "Mathematical Games: Circles and spheres, and how they kiss and pack", Scientific American, 218 (5): 130–139, doi:10.1038/scientificamerican0568-130, JSTOR 24926234
Lagarias, Jeffrey C.; Mallows, Colin L.; Wilks, Allan R. (2002), "Beyond the Descartes circle theorem", The American Mathematical Monthly, 109 (4): 338–361, arXiv:math/0101066, doi:10.2307/2695498, JSTOR 2695498, MR 1903421
Lachlan, R. (1886), "On Systems of Circles and Spheres", Philosophical Transactions of the Royal Society of London, 177: 481–625, JSTOR 109492; see "Spheres touching one another", pp. 585–587
Pedoe, Daniel (1967), "On a theorem in geometry", The American Mathematical Monthly, 74 (6): 627–640, doi:10.2307/2314247, JSTOR 2314247, MR 0215169
Mackenzie, Dana (January–February 2010), "A tisket, a tasket, an Apollonian gasket", Computing Science, American Scientist, vol. 98, no. 1, pp. 10–14, JSTOR 27859441, All of these reciprocals look a little bit extravagant, so the formula is usually simplified by writing it in terms of the curvatures or the bends of the circles.
Wilker, J. B. (1969), "Four proofs of a generalization of the Descartes circle theorem", The American Mathematical Monthly, 76 (3): 278–282, doi:10.2307/2316373, JSTOR 2316373, MR 0246207
Northshield, Sam (2014), "Complex Descartes circle theorem", The American Mathematical Monthly, 121 (10): 927–931, doi:10.4169/amer.math.monthly.121.10.927, hdl:1951/69912, JSTOR 10.4169/amer.math.monthly.121.10.927, MR 3295667, S2CID 16335704
Hajja, Mowaffaq (2009), "93.33 on a Morsel of Ross Honsberger", The Mathematical Gazette, 93 (527): 309–312, JSTOR 40378744
Bradley, Christopher J. (March 2003), "Heron triangles and touching circles", The Mathematical Gazette, 87 (508): 36–41, doi:10.1017/s0025557200172080, JSTOR 3620562, S2CID 125024700
McGonagle, Annmarie; Northshield, Sam (2014), "A new parameterization of Ford circles", Pi Mu Epsilon Journal, 13 (10): 637–643, JSTOR 24345283, MR 3235834
Veldkamp, G. R. (1985), "The Isoperimetric Point and the Point(s) of Equal Detour in a Triangle", The American Mathematical Monthly, 92 (8): 546–558, doi:10.1080/00029890.1985.11971677, JSTOR 2323159
Kocik, Jerzy (2007), A theorem on circle configurations, arXiv:0706.0372
Kocik, Jerzy (2010), "Golden window" (PDF), Mathematics Magazine, 83 (5): 384–390, JSTOR 10.4169/002557010x529815, doi:10.4169/002557010x529815

Kocik, Jerzy (2019), Proof of Descartes circle formula and its generalization clarified, arXiv:1910.09174

koreascience.kr

Takinami, Susumu; Michiwaki, Yoshimasa (1984), "On the Descartes circle theorem" (PDF), Journal for History of Mathematics, 1 (1), Korean Society for History of Mathematics: 1–8

proquest.com

search.proquest.com

Mackenzie, Dana (March–April 2023), "The princess and the philosopher", American Scientist, vol. 111, no. 2, pp. 80–84, ProQuest 2779946948

psu.edu

citeseerx.ist.psu.edu

A definition of stereographic distance can be found in Li, Hongbo; Hestenes, David; Rockwood, Alyn (2001), "Spherical conformal geometry with geometric algebra" (PDF), Geometric Computing with Clifford Algebras, Springer, pp. 61–75, CiteSeerX 10.1.1.412.4949, doi:10.1007/978-3-662-04621-0_3, ISBN 978-3-642-07442-4

semanticscholar.org

api.semanticscholar.org

Soddy, F. (June 1936), "The Kiss Precise", Nature, 137 (3477): 1021, Bibcode:1936Natur.137.1021S, doi:10.1038/1371021a0, S2CID 6012051
Hogendijk, Jan P. (1986), "Arabic traces of lost works of Apollonius", Archive for History of Exact Sciences, 35 (3): 187–253, doi:10.1007/BF00357307, JSTOR 41133783, MR 0851067, S2CID 121613986
Steiner, Jakob (January 1826), "Fortsetzung der geometrischen Betrachtungen (Heft 2, S. 161)", Journal für die reine und angewandte Mathematik, 1826 (1), pp. 252–288, fig. 2–25 taf. III, doi:10.1515/crll.1826.1.252, S2CID 121590578
Soddy, Frederick (December 1936), "The hexlet", Nature, 138 (3501): 958, Bibcode:1936Natur.138..958S, doi:10.1038/138958a0, S2CID 28170211
Levrie, Paul (2019), "A straightforward proof of Descartes's circle theorem", The Mathematical Intelligencer, 41 (3): 24–27, doi:10.1007/s00283-019-09883-x, hdl:10067/1621880151162165141, MR 3995314, S2CID 253818666
Northshield, Sam (2014), "Complex Descartes circle theorem", The American Mathematical Monthly, 121 (10): 927–931, doi:10.4169/amer.math.monthly.121.10.927, hdl:1951/69912, JSTOR 10.4169/amer.math.monthly.121.10.927, MR 3295667, S2CID 16335704
Tupan, Alexandru (2022), "On the complex Descartes circle theorem", The American Mathematical Monthly, 129 (9): 876–879, doi:10.1080/00029890.2022.2104084, MR 4499753, S2CID 251417228
This is a special case of a formula for the radii of circles in a Steiner chain with concentric inner and outer circles, given by Sheydvasser, Arseniy (2023), "3.1 Steiner's porism and 3.6 Steiner's porism revisited", Linear Fractional Transformations, Undergraduate Texts in Mathematics, Springer International Publishing, pp. 75–81, 99–101, doi:10.1007/978-3-031-25002-6, ISBN 978-3-031-25001-9, S2CID 258177153
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Eriksson, Nicholas; Lagarias, Jeffrey C. (2007), "Apollonian Circle Packings: Number Theory II. Spherical and Hyperbolic Packings", The Ramanujan Journal, 14 (3): 437–469, arXiv:math/0403296, doi:10.1007/s11139-007-9052-6, S2CID 14024662

siu.edu

lagrange.math.siu.edu

Kocik, Jerzy (2007), A theorem on circle configurations, arXiv:0706.0372
Kocik, Jerzy (2010), "Golden window" (PDF), Mathematics Magazine, 83 (5): 384–390, JSTOR 10.4169/002557010x529815, doi:10.4169/002557010x529815

Kocik, Jerzy (2019), Proof of Descartes circle formula and its generalization clarified, arXiv:1910.09174

srce.hr

hrcak.srce.hr

Garcia, Ronaldo; Reznik, Dan; Moses, Peter; Gheorghe, Liliana (2022), "Triads of conics associated with a triangle", KoG (26), Croatian Society for Geometry and Graphics: 16–32, arXiv:2112.15232, doi:10.31896/k.26.2, S2CID 245634505

tandfonline.com

Kocik, Jerzy (2007), A theorem on circle configurations, arXiv:0706.0372
Kocik, Jerzy (2010), "Golden window" (PDF), Mathematics Magazine, 83 (5): 384–390, JSTOR 10.4169/002557010x529815, doi:10.4169/002557010x529815

Kocik, Jerzy (2019), Proof of Descartes circle formula and its generalization clarified, arXiv:1910.09174

zbmath.org

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