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family of ratonal circles exists

Source: 2006 VMEO III Shortlist SL G5 Vietnamese Mathematics e - Olympiad https://artofproblemsolving.com/community/c2461015_vmeo__viet

October 29, 2021
geometrycircles

Problem Statement

Prove that there exists a family of rational circles with a distinct radius {(On)}\{(O_n)\} (n=1,2,3,...)(n = 1,2,3,...) satisfying the property that for all natural indices nn, circles (On)(O_n),(On+1)( O_{n+1}), (On+2)(O_{n+2}),(On+3)(O_{n+3}) are externally tangent like in the figure.
https://cdn.artofproblemsolving.com/attachments/b/f/5655e677e7c4f203b63afe82c50088e9ef97f5.png
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