Geometrical problems

To be used as a KWARC qualifying examination

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 * prove Pythagoras' theorem by construction
 * Application of Pythagoras: How far away is the horizon?
 * cut this into four equal (congruent) parts. Two and three would be simple.
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 * cut any rectangle into five equal parts. Two and four would be simple.
 * prove by construction that a square is the biggest rectangle with a given perimeter
 * Imagine an infinite plain colored with two colors (any regular or irregular distribution of colors). Show that you can always place a stick of a given length in such a way that both ends are placed on spots of the same color.