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).

(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

Bottema's Theorem

1. Bottema's Theorem
2. An Elementary Proof of Bottema's Theorem
3. Bottema's Theorem - Proof Without Words
4. On Bottema's Shoulders
5. On Bottema's Shoulders II
6. On Bottema's Shoulders with a Ladder
7. Friendly Kiepert's Perspectors
8. Bottema Shatters Japan's Seclusion
9. Rotations in Disguise
10. Four Hinged Squares
11. Four Hinged Squares, Solution with Complex Numbers
12. Pythagoras' from Bottema's
13. A Degenerate Case of Bottema's Configuration
14. Properties of Flank Triangles
15. Analytic Proof of Bottema's Theorem
16. Yet Another Generalization of Bottema's Theorem
17. Bottema with a Product of Rotations
18. Bottema with Similar Triangles
19. Bottema in Three Rotations
20. Bottema's Point Sibling

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