So you need a proof of Ptolemy's Theorem , right ?

Here's what i proved not long ago :

There are fours arcs , choose the biggest one , say Let be a point on the arc such that note that is an isosceles trapzeium so we have and . Then consider the area of the quadrilateral we find that it is equal to :

where is the intersection of the diagonals and is the included angle (acute one ) .

Since , we have

Also

But

Therefore,

By equating and we have

Here is another method ,

Let

Since