It looks as though FG is parallel to DE, but how easy is it to prove or refute this conjecture? It probably depends on one's familiarity with the family of circle theorems.

We start by viewing the diagram dynamically, using three movies showing progressively more extreme positions of F (and G).

During this process we refer to a circle theorem about cyclic quadrilaterals, and to a related theorem that applies when G is at B.

We then look at a theorem relating the lengths of various line segments. This theorem is based on the fact that the three triangles shown in the figure turn out to be similar. It is easily proved algebraically using ratio but we also try to make sense of it using area, which is how it is framed in Euclid's Elements.

On the next page we move F to D and back again (and G to E and back again).
Is FG parallel to DE ?

a high-res pdf file of this new GEOaa-zz task