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Problem of the Week
Problem E
Three Squares

The three squares \(ABCD\), \(AEFG\), and \(AHJK\) overlap as shown in the diagram.

Three squares of different sizes all share a vertex A as
follows. The largest square AHJK has point B on side AH, point D on
adjacent side AK, and point C in its interior forming the smallest
square ABCD. It also has point E on side AH (between B and H), point G
on side AK (between D and K), and point F in its integer forming the
middle square AEFG. The L-shaped region inside the middle square that is
not covered by the smallest square is shaded.

The side length of each square, in centimetres, is a positive integer. The area of square \(AEFG\) that is not covered by square \(ABCD\) is \(33\text{ cm}^2\). That is, the area of the shaded region \(BEFGDC\) is \(33\text{ cm}^2\). If \(DG = GK\), determine all possible side lengths of each square.