The semicircle above the distance AB cuts the middle perpendicular m of the distance AB in the point C. On the quarter arc between point A and C is chosen a point D and is drawn through him the tangent t to the semicircle. The distance BD cuts m in the point E. The parallel to AB through E cuts t in point F. On which line moves F if D moves on the quarter arc from A to C without reaching the point C?

F moves straight down to A, but why? How can I find a prove?

Sincerly, Enoimreh

First, let me restate the problem in standard mathematical English, to make it clearer to others:

A semicircle whose diameter is the segment AB cuts the perpendicular bisector m of segment AB in the point C. On arc AC, take any point D. Line t is tangent to the semicircle at D. Segment BD cuts line m in the point E. The parallel to AB through E cuts t in the point F. Find the locus (that is, path) of point F, as D moves on the arc from A to C without reaching the point C.

You have conjectured (possibly using geometry software) that the locus of F is a line through A perpendicular to AB, and want to prove it.

I would first draw segments MD and MF. Then I would mark angles that are congruent to one another, notably to angles ABD and AMD. These may be enough to suggest a proof; or you may find it helpful to make a further conjecture that MF and BD are parallel, and try to prove that as an intermediate step.

If you need more help, be sure to show what you have done so far; a marked-up copy of your picture may help.