14 Jun
A Chain of Circles in a Sangaku and a Generalization
A nice sangaku was listed by Fukagawa and Pedoe as Problem 1.3.3.
A sangaku from Fukagawa & Pedoe, 1.3.3
Points O1, O2, and O3 are collinear points and the circles O1(r), O2(r), and O3(r) touch each other, the first touching the second and the second the third. The circle O(R) circumscribes the three given circles, touching the first and the third internally. The chord PQ of this circle is an internal common tangent to the circles O1(r) and O3(r). Show that
PQ = R + 3r .
The problem is a simple exercise in the geometry of triangle that admits a nice generalization. Moreover, the configuration possesses a curious property that might have been overlooked by the Japanese author.
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