(1) Connect BD so that the vertical lines of BD intersect AB and AC at points E and F respectively. The drawing has been completed.

Let BF=y, then cf = 2-y.

Because cosB= 1/3, BC=2, BC is the base of isosceles triangle, BC=2, it is easy to get AB= 1*3=3.

Because d is the midpoint of AC, DC=3/2.

And BF = DF = y because BF and df are symmetrical about EF.

So y 2 = (2-y) 2+(3/2) 2-2 * (3/2) * (2-y) * (1/3) can be obtained from the cosine theorem in the triangle DFC.

The solution gives y= 17/ 12.

(2) y 2 = (2-y) 2+x 2-2 * x * (2-y) * (1/3) The formula obtained from the cosine theorem in (1).

Y = (3x2-4x+12)/(12-2x) .0 less than x less than or equal to 2. ..

(3) According to the similarity conditions, X=Y must be substituted into the equation in (2) to get x=2 or 1.2.

That is CD=2 or 1.2.

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