Ruler and Compass Construction 20
Doubling the Area of a Given Square

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1. Draw diagonal BD, which will be the side of the doubled square
2. Extend BD
3. Semi-circle centered at B of arbitrary radius
4. Arc centered at E of arbitrary radius
5. Arc centered at F of same radius
6. Ray BG, perpendicular to BD
7. Arc centered at B of radius BD
8. Arc centered at D, radius BD
9. Arc centered at H, radius BD
10. Segments DI, HI
Why it works: The area of the original square is AB^2, so we want a square of area 2AB^2. If the side of the new square is s, then we want s^2 = 2AB^2, or s = AB*sqrt(2), which is the length of the diagonal of the square.
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