Dudeney's dissection of a scalene triangle to form a square

This applet is inspired by H.E. Dudeney's problem of dissecting the triangle so as to form a square (see ivandewinne's posting: http://www.geogebratube.org/student/m352). Here, the triangle (in grey) need not be equilateral.
Move C about to explore the picture (B should remain at (1,0)). What are the restrictions on the position of C, so that the areas of the green square and triangle are equal? How can the square be constructed from the triangle? Hint: the semicircle might help to locate the points H & F.