My browser settings do not allow applets. So I haven't understood the inversion method given by Alex. Maybe i should begin studying the matter contained in this website (I am a new member and this is only my second visit to this site)
Here is a simple method to construct the diameter - (Refer to the attached file)

Plot another point on the other side of the center of the circle (mirror image of the point - about the center of the circle - at which the diameter subtends the given angle). Using the point and its mirror image, draw an arc that inscribes the supplementary angle of the angle that the diameter is supposed to subtend at the given point. The point of intersection of the arc and the circle gives one end of the required diameter.