Puzzle of the Week #304 - Compass Points

Four lines of lengths 12, 31, 27 and 10 are drawn respectively from the North, East, South and West points on a circle, heading directly away from the circle’s centre as shown.

compass points puz.JPG

 The four endpoints are joined with straight lines to form an irregular quadrilateral. Two opposite angles of this quadrilateral, A and B, add to 180 degrees.

 What is the radius, r, of the circle?