In a cycic quadrilateral ABCD, let the sides AB, BC, CD, DA be of lengths a, b, c, d, respectively.
If the diagonal AC and BD have lengths m and n, respectively, prove that ac + bd = mn
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