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Inversive Properties of the Plane n ‐Line, II: An Infinite Six‐Fold Chain of Circle Theorems
Author(s) -
Longuet-Higgins M. S.,
Parry C. F.
Publication year - 1979
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-19.3.541
Subject(s) - mathematics , combinatorics , diagonal , table (database) , line (geometry) , geometry , computer science , data mining
It is shown that the elements of the configurations CLn and DLn occurring in the circle chains of Clifford and de Longchamps respectively may be arranged in a Table so that the elements of CLn occupy the first column, and the elements of DLn occupy three adjacent diagonals. By using the results on conjugate pentads derived in an earlier paper, it is shown that there exists a new chain of circle theorems, occupying the first six rows of the Table, and extending indefinitely in either direction. The elements in each column form a CL5 , and alternate columns are homothetically related to the centre Ω of the Miquel circle of the original CL5 . Among other results it is proved that if G is a general CL5 , then there exists a unique pair of points T and T ′ such that the images of T in the sixteen circles of G are the sixteen points of another CL5 say G ′; and the images of T ′ in the sixteen circles of G ′ are the points of G .