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 Two squares in a circle (Posted on 2014-12-05)

I drew a square of side 5 together with its circumcircle. At the middle of the arc between a side and the circle, I drew a smaller square with two vertices on the side, and the other two on the circle.

What was the area of the smaller square?

 Submitted by broll Rating: 3.0000 (2 votes) Solution: (Hide) Idea: there is only one small square that will meet the constraints of the problem, so it needs only be constructed to produce the desired answer. Divide the 5*5 square ABCD into 25 little 1*1 squares (1,1) to (5,5) and construct semi-diagonal DO from D to the centre O of the circumcircle, S. DO is also a diagonal of (1,1), (2,2) and (3,3), the last of these being centred on O. Construct the other diagonal of (1,1) and extend it to intersect S at H. Do the same with little square (2,2) to intersect CD at F. Lastly, at the common vertex of (1,1) and (2,2) construct a line, parallel to the two diagonals just drawn, intersecting CD at E and S at G. Clearly quadrilateral EFGH is a square, and is of the same size as (3,3), so its area is exactly 1.

 Subject Author Date Solution Dej Mar 2014-12-06 06:46:01 solution Ady TZIDON 2014-12-06 01:38:09 re: solution Ady TZIDON 2014-12-06 01:05:44 solution Charlie 2014-12-05 15:20:18

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