The first thing you have to keep in mind is the symmetry of the problem. Here you have an equiliateral triangle, meaning all sides have the same length and all angles measure the same. That gives you the first clue, which is that the angles of the triangle are all 60 degrees (360/3).
The second thing you have to realize is that any line from a vertex to the circumcenter will bisect the angle of 60 degrees into two 30 degree angles. Put this together with the fact that a line that goes through the midpoint of one of the sides of the triangle and the circumcenter.
If you draw all this out you will find out that you wind up with a nice set up to the problem, where you are looking for the length of one of the sides of a right triangle.
The final step is to use trigonometry. You'll have a right triangle with an angle of 30 degrees, and a hypotenuse equal in length to the distance between the circumcenter and the vertex. Solve for the side they are asking about (think about the sine and cosine definitions), and you should be set.