7 = 2 + 5 Sangaku
A slew of sangaku problems deal with chains of inscribed circles, see, for example, Steiner's sangaku. The elegant sangaku below is the simplest in the collection by Fukagawa and Pedoe (1.8.6), yet it gives a chance to discuss a formula that was not used so far anywhere else at the site.
The points T, A, B are collinear and TA = 2r and TB = 2s. The circles C_{1}(r) and C_{2}(s) have diameters TA and TB respectively. The circle O_{1}(r_{1}) touches AB, touches C_{1}(r) externally and C_{2}(s) internally, and we then form a chain of contact circles O_{i}(r_{i})
7 / r_{4} = 2 / r_{7} + 5 / r_{1}.
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
Contact Front page Contents Geometry Up
Copyright © 19962018 Alexander Bogomolny
This sangaku dates from 1814 and was written in the Gumma prefecture.
The points T, A, B are collinear and TA = 2r and TB = 2s. The circles C_{1}(r) and C_{2}(s) have diameters TA and TB respectively. The circle O_{1}(r_{1}) touches AB, touches C_{1}(r) externally and C_{2}(s) internally, and we then form a chain of contact circles O_{i}(r_{i})
7 / r_{4} = 2 / r_{7} + 5 / r_{1}.
Solution
The derivation of a more traditional formula for the common chain of circles inscribed in an arbelos
r_{n} = ts / (n² + t + t²).
is easily adapted to the present case. The result is the following formula
(*)  r_{n} = 4ts / ((2n  1)² + 4t + 4t²), 
where t = r / (s  r). Denoting
 r_{1} = 4s / (1 + S),
 r_{4} = 4s / (7² + S),
 r_{7} = 4s / (13² + S),
which we plug into the required identity
7 / r_{4} = 2 / r_{7} + 5 / r_{1}
getting
7(7² + S) = 2(13² + S) + 5(1 + S)
which simplifies to
7·7² = 2·13² + 5.
As one can easily verify, both sides are equal to 343.
Sangaku

[an error occurred while processing this directive]

[an error occurred while processing this directive]
Contact Front page Contents Geometry Up
Copyright © 19962018 Alexander Bogomolny