Show square inscribed in circle has maximum area

[t_n_p]

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?

 

[Borek]

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.

 

[t_n_p]

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?

 

[rishicomplex]

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

 

[t_n_p]

that gives me the area, but it doesn't prove that it is the maximum area for a 4 sided shape.

 

[rishicomplex]

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