Here is another elegant Quantum math magazine Brainteaser problem.
“Two squares are inscribed in a semicircle as shown in the figure at left. Prove that the area of the big square is four times that of the small one.”
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Let the big square ABCD clockwise with A at the top left corner, the points A and B on the circumference and the points C and D on the diameter.
The small square CEFG clockwise with the point E on BC, the point F on the circumference and the point G on the diameter.
Let O be the centre of the semicircle. Join AO, BO and FO.
The right angled triangles AOD and BOC are congruent as AO=BO= radius and AD=BC. Hence DO=OC= 1 unit say. Thus the side of the big square is 2 units and the area is 4 square units.
The radius will be √5.
Let CG, the side of the small square be ‘a’ units. In the right angled triangle OFG, OF=√5, FG=a and OG=(1+a). Using Pythagorean, and solving for ‘a’, we get a=1 unit.
This gives small square area as 1 square unit, and the ratio as 4:1.