Show square inscribed in circle has maximum area

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Homework Help Overview

The problem involves demonstrating that a square inscribed in a circle has the largest area among all four-sided polygons. The original poster expresses confusion about how proving that the sides are equal relates to showing that the square has maximum area.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss expressing the area of an inscribed quadrilateral as a function of its side lengths, with some suggesting that starting with a rectangle may simplify the problem. There is also mention of using coordinate geometry and parametric equations to derive the area.

Discussion Status

Some participants have offered guidance on using specific mathematical approaches, such as derivatives and trigonometric formulas, to explore the area of the quadrilateral. However, there is no explicit consensus on the best method to demonstrate that the square has the maximum area.

Contextual Notes

Participants are navigating the complexities of proving maximum area under the constraints of inscribed shapes and are considering the implications of angles and side lengths in their reasoning.

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Homework Statement


Show that the square, when inscribed in a circle, has the largest area of all the 4-sided polygons. Try to show that all sides of a quadrilateral of maximal area have to be of
equal length.

The Attempt at a Solution


How do you start?
I don't get how showing the sides are equal will help prove that the square has maximal area.
Any hints?
 
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Try to express surface of the inscribed 4-sided polygon as a function of the length of its sides. General case will be more difficult, but for a specific case of rectangle it should be relatively easy to express lengths as a function of one variable.

Once you have that, find maximum.
 
Borek said:
Try to express surface of the inscribed 4-sided polygon as a function of the length of its sides. General case will be more difficult, but for a specific case of rectangle it should be relatively easy to express lengths as a function of one variable.

Once you have that, find maximum.

I don't quite understand.
Say I have a square, let the side length = a.

area = a^2.
Then maximize by taking derivative?
 
Real easy to prove with coordinate geometry, but you need to know the parametric equation of a circle and the area of a quadrilateral.

Any point on a circle (with origin as centre and radius r) is (r cosx, r sinx)
and area of a quadrilateral with vertices (x1, y1), (x2, y2), (x3, y3), (x4, y4),
is #8 here http://www.mathisfunforum.com/viewtopic.php?id=3301

take four points, put them in the formula, the result becomes obvious
 
that gives me the area, but it doesn't prove that it is the maximum area for a 4 sided shape.
 
it does! the result for area of a general quadrilateral in a circle comes to be (assuming radius r and angles A, B, C, D
r2/2[sin(A-B) + sin(B-C) + sin(C-D) + sin(D-A)]
the maximum value of which (2r2) is attained only when each of the angular differences is 90 degrees which results in a square

did u try solving it? you need basic trignometric formulae
 

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