Ruler and Compass Construction 24
Square with Same Area as Given Rectangle

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1. Extend ray AB
2. Arc centered at B, radius BC
3. Arcs centered at A, arbitrary radius
4. Arcs centered at E, same radius
5. Segment FG, so H is midpoint of AE
6. Semi-circle centered at H, through A
7. Extend ray BC
8. Intersection I; IB is the side of the square
9. Arc centered at B, radius BI
10. Arc centered at J, radius BI
11. Arc centered at I, radius IB
12. Square HJKI
13. Right triangles show IB/AB = BE/IB

Why it works:
Draw chords AI and IE. Then angle AIE is inscribed in a semicircle, so angle AIE is a right angle.
Then triangle AIB is similar to triangle ABI, both having angle A and a right angle, and likewise
triangle AIB is similar to triangle IBE. Thus ABI is similar to IBE, and so AB/BI = BI/BE.
Cross-multiplying then gives AB*BE = BI*BI. Since BE = BC, we have area of rectangle = AB*BC = BI*BI = area
of square.