Two circles are externally tangent and have a common external tangent. A square is placed in the area between the two circles and the line in such a way that it is touching all three objects. The main question is what the length of the side of the minimal square is. The described problem originates in Japan and is called Morikawa's problem. This master's thesis presents all the possible positions of the square, why the minimal square even exists, and geometric arguments for whether or not the minimal square can exist in a given position. An explicit formula for
a function whose minimum value on a certain interval is exactly the length of the square's shortest side is also given. This length can be expressed in terms of a root of a certain polynomial of degree 10. Finally, with the help of Galois theory, we proved that the polynomial of degree 10 cannot be solved by radicals and why that is, as well as why the length of the minimal square's side as a function of radius is not a radical function.
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