Let AB=BC=x, then AC=xsqrt(2) (Pythagoras).
Area of equilateral triangle of side a: (1/2)a^2sqrt(3)/2 because height is asqrt(3)/2 (that is, asin60).
Area of BCD=x^2sqrt(3)/4; area of ACE=(xsqrt(2))^2sqrt(3)/4=2x^2sqrt(3)/4.
Therefore area of ACE=twice the area of BCD.
An easier approach is to use the fact that if we know one linear ratio of two similar figures, then the ratio of their areas is the square of the linear ratio. The two equilateral triangles are similar and the ratio of their sides is AB:AC or BC:AC=1:sqrt(2) then the ratio of their areas is 1:2.