Method 1: AB/sin∠C=BC/sin∠A=AC/sin∠B=2R is used in the conclusion triangle ABC.

Then ∵∠DBC=60 degrees.

ACB = 30 degrees

∴∠BAC= 120 degree

∴ the diameter is 2R=BC/sin∠A=4/sin∠ 120 degrees =4/√3/2=8√3/3.

Method 2: It can be proved from an example in the textbook that the product of two sides of a triangle is equal to the product of the height of the third side and the diameter of the circumscribed circle.

Then in RT triangle BCD

Bc = 4 < DBC = 60 degrees

∴BD=2

In the RT triangle ABD

Available AB=4√3/3

∴AB×BC=BD×2R

That is, 4√3/3×4=2×2R.

2R=8√3/3

I wish you progress in your studies.