Ruler and Compass Construction 21
Circle of Twice the Area of a Given Circle

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1. Diameter AB
2. Arc centered at A, arbitrary radius
3. Arc centered at B, same radius
4. Segment OC, perpendicular to AB
5. Segment DB, length sqrt(2)*OB
6. Circle centered at D through B

Why it works:
The area of the original circle is pi*R^2, so we want a circle of area 2*pi*R^2. If the radius
of the new circle is s, then we want pi*s^2 = 2*pi*R^2, or s = R*sqrt(2), which is the length
of the hypotenuse of two radii at right angles..