# Four Hinged Squares

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

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*) - i*h* = -i(2*h* - *c*)

*s* - *k* = (*s* - *e*) + (*e* - *k*) = -i(*a* - i*h*) - *h* = -*a*i - 2*h*

*j* - *k* = -i(*s* - *k*) so that *h* = (*c* - *a*i)/4.

Replacing value of *h* in *j* - *k*, we get *j* - *k* = (i/2)(*c* + *a*i) = 2i*h*', (where *h*' denotes the conjugate of *h*) so that

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

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

### Bottema's Theorem

- Bottema's Theorem
- An Elementary Proof of Bottema's Theorem
- Bottema's Theorem - Proof Without Words
- On Bottema's Shoulders
- On Bottema's Shoulders II
- On Bottema's Shoulders with a Ladder
- Friendly Kiepert's Perspectors
- Bottema Shatters Japan's Seclusion
- Rotations in Disguise
- Four Hinged Squares
- Four Hinged Squares, Solution with Complex Numbers
- Pythagoras' from Bottema's
- A Degenerate Case of Bottema's Configuration
- Properties of Flank Triangles
- Analytic Proof of Bottema's Theorem
- Yet Another Generalization of Bottema's Theorem
- Bottema with a Product of Rotations
- Bottema with Similar Triangles
- Bottema in Three Rotations
- Bottema's Point Sibling

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