Dido's Problem/Variant 1

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Consider a frame consisting of $4$ rods freely hinged at their ends:


When will the area enclosed by the frame be a maximum?


When the quadrilateral formed by the frame is cyclic.

Proof 1

Let the frame be adjusted so that the vertices of the quadrilateral it forms lie on the circumference of a circle.

Suppose the $4$ corresponding arcs that form the circle are also hinged, like the frame, at the vertices.

Suppose the shape of the frame be changed by bending it at the hinges.

Then the arcs no longer form a circle.

Thus the arcs no longer enclose the maximum area.

But the areas of the segments formed by the arcs and the sides of the frame are the same.

So the area enclosed by the frame is not as great as it was when the frame formed a cyclic quadrilateral.

Hence the result.


Proof 2

This problem is a direct application of the result:

Area of Quadrilateral with Given Sides is Greatest when Quadrilateral is Cyclic