Find whether among all quadrilaterals, whose interiors lie inside a semi-circle of radius $r$, there exist one (or more) with maximum area. If so, determine their shape and area.
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The existence of a maximum is warranted by a compactness argument.
Trivially, a quadrilateral achieving maximum area must be convex and have all its vertices on the boundary of the semi-circle.
We may assume two vertices lie on the diameter of the semi-circle, otherwise consider the side with least distance from the center of the circle of radius $ r$ and prolong it to a full chord; then the quadrilateral is also inscribed in the semi-circle with diameter this chord $ \leq 2r$.
Now symmetrize with respect with the diameter. The result is a polygon with $ 6$ vertices, of double the area, contained in the circle of radius $ r$.
By a celebrated result of Steiner, the largest area of an $ n$-gon contained in a circle is realized by the regular $ n$-gon, therefore in our case the maximum area is given by a half regular hexagon.