In our original task, we can use knowledge about given circles (and cyclic quadrilaterals) to deduce something about angles (to show that certain lines are parallel). Here we use knowledge about given angles to deduce something about the existence (or not) of circles.

ABC is 'any' triangle.
For each diagram, can we draw a circle through B, C, D and E
• always, whatever the triangle's shape
• never, regardless of the triangle's shape
• sometimes, when the triangle has a specific kind of shape?

the end

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