🚀 go-pugleaf

In article <5hs7fa$4p7@opus.vcn.bc.ca>
          landmark@vcn.bc.ca (Scott Phung) wrote:

->Brian McCague <brian@mccague.demon.co.uk> wrote:
->BM> In article <5hnv2g$95s@opus.vcn.bc.ca>, Scott Phung <landmark@vcn.bc.ca>
->BM> writes
->BM> >Here's a math problem that's been on my mind:
->BM> >
->BM> >  Suppose I have a circle with a radius r.  Then into it is
->BM> >inscribed a cyclic quadrilateral.  Is the area of this cyclic
->BM> >quadrilateral = 2r^2?  I know that if the cyclic quadrilateral
->BM> >was a square, then the area of the square = 2r^2.
->BM> >
->BM> >  More Generally, if a polygon is inscribed into a circle with
->BM> >a radius r, would the area of the polygon inscribed = area of
->BM> >regular polygon inscribed?
->BM> >
->BM> >  Also, does the center of the circle lie on any of the two
->BM> >diagonals of the cyclic quadrilateral?
->BM> >
->BM> >  Can anyone confirm the any of above?
->BM> >
->BM> >
->BM> Consider a quadrilateral formed by four points that are very close
->BM> together on the circle's circumference.  It follows that the answer to
->BM> the above questions is 'no'.
->BM> --
->BM> Brian McCague
->
->But does the angles diagonally opposite of each other add to 180?
->
->A ......... D
->   .     .
-> B  ..... C
->
->Does <B + <D = 180
-> and <A + <C = 180?
->
->Yes, it's easy to image four points close together, but do these
->four points satisfy the above?  Perhaps you can provide an
->example.
->
->If it adds to 180, my mistake and I will be beleive that
->this area is smaller than that of a inscribed square.

OK. Create a trapezium ABCD with angles (in degrees) 2, 178, 178, and
2, in that order. Make AD 5 cm long, and AB = CD = 2cm. It has a
rather pleasing line of symmetry, but that isn't important here. It is
cyclic by construction, by the converse of the cyclic quadrilateral
theorem(1). When you draw this circle you will see that with the
dimensions chosen the area of the circle will be very large compared
to the area of the area of the trapezium. Precise calculation is left
as an exercise to the reader. :-)

In fact, the inscribed regular polygon with n sides has the maximum
area of all inscribed n-sided polygons.

(1) If the opposite angles of a quadrilateral are supplementary, the
quadrilateral is cyclic.

-- 
Malcolm in sunny Berkhamsted