In the figure, each side of square ABCD has length 1, the length of line segment CE is 1, and the length of line segment BE is equal to the length of line segment DE.  What is the area of the triangular region BCE?

~$\frac{1}{3}$~

~$\frac{\sqrt{2}}{4}$~

~$\frac{1}{2}$~

~$\frac{\sqrt{2}}{2}$~

~$\frac{3}{4}$~