Strandbeest & other tasks

Theo Jansen is the man who reinvented the wheel by constructing "strandbeests" as seen here on TED.

Construct a GeoGebra Strandbeest ↓

... or find points in triangles

Start with two triangles. Use some tools (which tools?) to do this ↑

The blue intersection points are not the same in the two different cases.

...or do the following

In the worksheet above, the segments are of equal length, OA=AB=BC=CD=DE=EF.

Call the angle AOB α. If α is sufficiently small, there is an angle DFE, the yellow angle in the applet above.

Find an expression for the angle DFE in terms of α!

How large can the angle α be without DFE disappearing? Why?