A hiking club wants to hike around a lake along an exactly circular route. On the shoreline they determine two points, which are the most distant from each other, and start to walk along the circle, which has these two points as the endpoints of its diameter. Can they be sure that, independent of the shape of the lake, they do not have to swim across the lake on any part of their route?
Solution
Suppose the shape of the lake is an equilateral triangle. Then the two points which are the most distant from each other are two vertices of the triangle. The circle, which has these two points as the endpoints of its diameter, does not cover the whole triangle, because the distance of the third vertex from the center of the circle is of the length of the side of the triangle, but the radius of the circle is only of this length.
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