If a parallelogram has all its sides equal and one of its diagonal is equal to a side, show that its diagonals are in the ratio : 1.

Given: ABCD is a parallelogram, where AC and BD are the diagonals meeting at O. AB = BC = AC. 
To Prove: BD : AC ::  : 1
Proof: In Δ ABC, AB = BC = CA (given).
                                   = a (say)
Hence ABC is an equilateral triangle. (Definition of equilateral triangle)
AC and BD are the diagonals of parallelogram ABCD,
⇒ AC = BD (Diagonals of a parallelogram bisect each other)
or AO = OC.
i.e BO is the median of the equilateral ABC.
Hence BO =a
∴ BD = a   
⇒ BD : AC :: a : a
⇒ BD : AC ::  : 1.