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
By equating and we have
Here is another method ,