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:

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).

References

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

Bottema's Theorem

1. Bottema's Theorem
2. An Elementary Proof of Bottema's Theorem
3. On Bottema's Shoulders
4. On Bottema's Shoulders II
5. Friendly Kiepert's Perspectors
6. Bottema Shatters Japan's Seclusion
7. Rotations in Disguise
8. Four Hinged Squares
9. Four Hinged Squares, Solution with Complex Numbers
10. Pythagoras' from Bottema's

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