In a cycic quadrilateralABCD, let the sidesAB, BC, CD, DAbe of lengthsa, b, c, d,respectively.

If the diagonalACandBDhave lengthsmandn, respectively, prove thatac+bd = mn

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- Jun 9th 2010, 07:24 PMcedriccPtolemy's Theorem
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* - Jun 9th 2010, 09:13 PMsimplependulum
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

- Jun 18th 2010, 03:14 AMcedricc
LOL, uhmm, wheres all the numbers gone after this website upgrade?? can you rewrite this again?? sorry! thanks!