I have a result, but I hope for you that there is a more simple and elegant way to do the question. Here is my method:

1. Use triangle ABC to calculate the length of the radius:

2. You'll get 4 isosceles triangles with the radius as their equal legs. Calculate the interior angles of the 3 known isosceles triangles at the center of the circle. the 4th angle add up to 360°. I've got

3. Calculate the length of

I've got AD = 3.