809:
20:
1009:
into two parallel lines, and the circles of the Pappus chain into a stack of equally sized circles sandwiched between the two parallel lines. Hence, the points of tangency between the transformed circles lie on a line midway between the two parallel lines. Undoing the inversion in the circle, this
627:
327:
692:
413:
424:
816:, the four initial circles of the Pappus chain are transformed into a stack of four equally sized circles, sandwiched between two parallel lines. This accounts for the height formula
799:
178:
990:
The same inversion can be used to show that the points where the circles of the Pappus chain are tangent to one another lie on a common circle. As noted above, the
896:
circle; hence, the other circles of the Pappus chain are transformed into similarly sandwiched circles of the same diameter. The initial circle
639:
360:
701:
1099:
339:, the centers of the two circles that define the arbelos; these points correspond to the midpoints of the line segments
1126:
1068:
1018:
In these properties of having centers on an ellipse and tangencies on a circle, the Pappus chain is analogous to the
1183:
1161:
1094:(reprint of 1929 edition by Houghton Mifflin ed.). New York: Dover Publications. pp. 116–117.
110:. The Pappus chain consists of the circles in the shaded grey region, which are externally tangent to
622:{\displaystyle (x_{n},y_{n})=\left({\frac {r(1+r)}{2}}~,~{\frac {nr(1-r)}{n^{2}(1-r)^{2}+r}}\right)}
1092:
Advanced
Euclidean Geometry: An elementary treatise on the geometry of the triangle and the circle
1188:
322:{\displaystyle {\overline {P_{n}U}}+{\overline {P_{n}V}}=(r_{U}+r_{n})+(r_{V}-r_{n})=r_{U}+r_{V}}
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1054:
1118:
1111:
44:
8:
991:
1122:
1095:
1064:
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1178:
863:
332:
1145:
40:
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1019:
1078:
161:
All the centers of the circles in the Pappus chain are located on a common
832:
and the fact that the original points of tangency lie on a common circle.
808:
892:, are transformed into parallel lines tangent to and sandwiching the
124:(the outer circle). Let the radius, diameter and center point of the
1150:
19:
28:
162:
60:
165:, for the following reason. The sum of the distances from the
106:, respectively, and let their respective centers be the points
36:
1022:, in which finitely many circles are tangent to two circles.
878:
circle is transformed into itself. The two arbelos circles,
687:{\displaystyle r={\tfrac {\overline {AC}}{\overline {AB}}},}
408:{\displaystyle r={\tfrac {\overline {AC}}{\overline {AB}}},}
971:. Adding these contributions together yields the equation
1113:
The
Penguin Dictionary of Curious and Interesting Geometry
1085:. Boston: Prindle, Weber, & Schmidt. pp. 112–118.
1081:(1981). "How did Pappus do it?". In Klarner, D. A. (ed.).
1010:
line of tangent points is transformed back into a circle.
48:
870:. The circle of inversion is chosen to intersect the
650:
371:
1142:
704:
642:
427:
363:
181:
95:. Let the radii of these two circles be denoted as
1110:
793:
686:
621:
407:
321:
1170:
169:circle of the Pappus chain to the two centers
117:(the inner circle) and internally tangent to
16:Ring of circles between two tangent circles
812:Under a particular inversion centered on
794:{\displaystyle r_{n}={\frac {(1-r)r}{2}}}
173:of the arbelos circles equals a constant
151:
128:circle of the Pappus chain be denoted as
1144:Floer van Lamoen and Eric W. Weisstein.
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18:
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1117:. New York: Penguin Books. pp.
874:circle perpendicularly, so that the
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77:, which are tangent at the point
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998:transforms the arbelos circles
847:circle above the base diameter
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866:centered on the tangent point
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1:
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7:
698:th circle in the chain is:
419:th circle in the chain is:
63:is defined by two circles,
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156:
1083:The Mathematical Gardner
1037:Ogilvy, pp. 54–55.
862:. This may be shown by
1090:Johnson, R. A. (1960).
694:then the radius of the
415:then the center of the
1056:Excursions in Geometry
943:, whereas the circles
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795:
688:
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152:Centers of the circles
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905:and the final circle
864:inverting in a circle
843:of the center of the
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632:Radii of the circles
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335:of this ellipse are
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45:Pappus of Alexandria
47:in the 3rd century
1184:Inversive geometry
1109:Wells, D. (1991).
1059:. Dover. pp.
994:centered at point
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1101:978-0-486-46237-0
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43:investigated by
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1189:Circle packing
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1160:Tan, Stephen.
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1146:"Pappus Chain"
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1137:External links
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23:A Pappus chain
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1070:0-486-26530-7
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1051:Ogilvy, C. S.
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1021:
1020:Steiner chain
1014:Steiner chain
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35:is a ring of
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1044:Bibliography
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55:Construction
39:between two
33:Pappus chain
32:
26:
1079:Bankoff, L.
836:The height
353:Coordinates
1173:Categories
1026:References
331:Thus, the
147:Properties
81:and where
1162:"Arbelos"
1151:MathWorld
992:inversion
764:−
728:−
675:¯
662:¯
590:−
563:−
511:−
396:¯
383:¯
278:−
225:¯
200:¯
1053:(1990).
961:−1
29:geometry
1179:Arbelos
927:
915:
851:equals
163:ellipse
157:Ellipse
61:arbelos
37:circles
1125:
1098:
1067:
855:times
545:
539:
31:, the
1061:54–55
1123:ISBN
1096:ISBN
1065:ISBN
885:and
337:U, V
333:foci
171:U, V
108:U, V
70:and
59:The
1119:5–6
1004:, C
952:to
849:ACB
636:If
357:If
138:, P
134:, d
101:, r
27:In
1175::
1148:.
1121:.
1063:.
987:.
981:nd
979:=
826:nd
824:=
346:AC
344:,
342:AB
51:.
49:AD
1164:.
1154:.
1131:.
1104:.
1073:.
1006:V
1002:U
1000:C
996:A
983:n
976:n
974:h
968:n
966:d
959:n
955:C
949:1
946:C
940:n
938:h
932:n
930:d
924:2
921:/
918:1
909:n
907:C
902:0
899:C
894:n
889:V
887:C
882:U
880:C
876:n
872:n
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859:n
857:d
853:n
845:n
840:n
838:h
828:n
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814:A
786:]
783:r
780:+
775:2
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767:r
761:1
758:(
753:2
749:n
745:[
742:2
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734:)
731:r
725:1
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716:=
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707:r
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671:B
668:A
658:C
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647:=
644:r
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593:r
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579:2
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442:,
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429:(
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403:,
392:B
389:A
379:C
376:A
368:=
365:r
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302:U
298:r
294:=
291:)
286:n
282:r
273:V
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265:(
262:+
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126:n
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119:C
114:U
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103:V
99:U
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90:C
85:U
83:C
79:A
74:V
72:C
67:U
65:C
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