Archimedes' Twins in the Edo Period

It is often said the Japanese mathematics during the Edo period of isolation has been developing independently of the events in the rest of the world. The sangaku below is included in the collection by Fukagawa and Pedoe as #1.5.6. The sangaku was written in 1838 on a surviving tablet in the Nagano prefecture. Without doubt, it relates to a problem discussed some 2000 years previously.

AB is a diameter of a given circle O(r) and CD is a perpendicular chord, the intersection is P. C1(t1) has diameter PA and C2(t2) has diameter PB. The circles O1(r1) and O'1(r1) touch CD, touch C1(t1) externally and O(r) internally, and the circles O2(r2) and O'2(r2) touch CD on the other side, touch C2(t2) externally and O(r) internally. Show that

r1 = r2.

Solution

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Copyright © 1996-2017 Alexander Bogomolny

The diagram depicts a double picture of Archimedes' twins turned 90°.

Sangaku

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Copyright © 1996-2017 Alexander Bogomolny

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