Four Hinged Squares

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

sangaku with four hinged squares

Four squares are hinged as shown. When points A, B, C are collinear, what is the relationship between the sides of squares BEKH and KINS?

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:

j - k = (j - h) + (h - k) = i(c - h) - ih = -i(2h - c)
s - k = (s - e) + (e - k) = -i(a - ih) - h = -ai - 2h
j - k = -i(s - k) so that h = (c - ai)/4.

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).

(Note: the problem we just discussed serves an example of application of complex numbers in geometry. There are many more.)

References

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