Four Hinged Squares

The problem of four hinged squares appeared in 1826 as a sangaku hung by Ikeda Sadakazu in an Azabu shrine, Tokyo.

Several solutions are available at this site: one makes use of Bottema's theorem, another comes from a wonderful book by Rothman and Fukagawa.

Below is a solution by Michel Cabart that employs complex numbers.

Let a, c, e, h, k, j be the complex numbers corresponding to points A, C, E, H, K, I, with the origin being in point B. Then:

Replacing value of h in j - k, we get j - k = (i/2)(c + ai) = 2ih', (where h' denotes the conjugate of h) so that IK = 2BH.

This solution yields an additional result: the axis of symmetry of lines (IK) and (BH) is inclined by 45° on line (BC).

References

H. Fukagawa, A. Rothman, Sacred Geometry: Japanese Temple Geometry, Princeton University Press, 2008, p. 149

Copyright © 1996-2009 Alexander Bogomolny