I could offer an additional piece of information, however I don't think it is necessary and possibly it adds no useful information. It is also not very easy to explain:

Code:

A
.
|..
|. ..
| . . .
| . . .
| . . .
| . . .
| . . .
|____...........
E D,,,,C,,,,B
,,,,,,,,,
,,,,,
F ,,

The area filled with comma signs represent a half circle.

Now we also know that line BD is the diameter of a circle on a plane which is at a right angle to the line AE. So we know that the radius R = BD/2. The center of the circle lies somewhere on the line CB. Note that C is not the center of the circle.

Point F lies on the perimeter of the circle. From C extends a line to F. This line is of course shorter than the radius. CF is at a right angle to BD.

The meaning of point F, as it relates to point C, is that we know that the angle CAF is the greatest angle which could be formed between line BD and the perimeter of the circle, such that the base (corresponding to CF) is at a right angle at line BD.

If point A was straight above the center of the circle, the angle CAF would be maximized iff C was the center of the circle and CF = radius. But as the case now is, point A has moved to the side and this CAF is maximized when C is offset from the center (and F must follow so that line CF remains at a right angle to line BD).