--- In

Hyacinthos@yahoogroups.com, "pskssvsk2003" <pskssvsk2003@y...>

wrote:

> Perhaps this is obvious to many of you;

>

> Can you please suggest how to construct a cyclic quadrilateral

given

> the lengths of its four sides?

> Is there a solution which does not first arithmetically calculate

the

> length of its radius using for example, Brahmagupta's formula?

>

> Sriram Panchapakesan

This problem is owned by the mathematician Sturm who has borned in

Genova in 1803.

Suppose the problem has been solved, draw a cyclic quadrilateral ABCD

and its' circumcircle. We know A+C=B+D=180. Draw the diagonal AC.Then

take a point on [CD (outside the circle) so that m(BAC)=m(DAE). Since

m(ABC)=m(ADE), then the triangles ABC and ADE are similar. a/d =

b/|DE| = |AC|/|AE| so |DE| = bd/a. Since we know |DE|, we also know

|CE| now. |AC|/|AE|=a/d=constant means that A is a point such that

the ratio of the distances to C and E is constant. So it must be on

the Apollonius circle. I think you can easily construct it now.

Mustafa