Hello, GAdams!

This is the original rectangle: Code:

* - - - - - - - - - *
| |
| |
1 | | 1
| |
| |
* - - - - - - - - - *
√2

The ratio of length-to-width is: .

Bisect it: Code:

* - - - - * - - - - *
| | |
| | |
1 | | | 1
| | |
| | |
* - - - - * - - - - *
½√2 ½√2

Each small rectangle looks like this: Code:

* - - - - - *
| |
| | ½√2
| |
* - - - - - *
1

The ratio of length-to-width is: .

And we must show that the two ratios are equal.

The first ratio is: .

The second ratio is: .

. . Rationalize: .

Therefore, the ratios are equal.