Sangaku à la V. Thébault: What Is This About?
A Mathematical Droodle
What if applet does not run? 
Sangaku

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Copyright © 19962018 Alexander Bogomolny
The applet is hopefully suggestive of a generalization of the following sangaku [Fukagawa & Pedoe, Example 2.4]:
O(R) is the circumcircle and I(r) is the incircle of triangle ABC, K is the foot of the perpendicular from C to AB. The circle P(r_{1}) touches CK and AB and touches O(R) internally, and Q(r_{2}) touches CK and AB and touches O(R) internally. Show that
r_{1} + r_{2} = 2r.
The problem has been written on a surviving tablet in 1901, in the Fukusima prefecture.
What if applet does not run? 
The applet suggests more than claimed: in fact I is the midpoint of the segment PQ. (Since all three circles touch the segment AB, the latter fact implies
The proof follows along the lines of another sangaku.
References
 H. Fukagawa, D. Pedoe, Japanese Temple Geometry Problems, The Charles Babbage Research Center, Winnipeg, 1989
Write to:
Charles Babbage Research Center
P.O. Box 272, St. Norbert Postal Station
Winnipeg, MB
Canada R3V 1L6
Activities Contact Front page Contents Geometry
Copyright © 19962018 Alexander Bogomolny