Dear Eileen Not sure whether you had a chance to investigate my "intuitive" guesses regarding cyclic & circum quads (quad circumscribed around a circle) I mentioned earlier. I just quickly checked it on Sketchpad and found the following:
(1) If the point of intersection of the diagonals of a cyclic quad is reflected over its sides, the resulting quad is a circum quad (ie. its angle bisectors are concurrent at the intersection of the diagonals of the cyclic quad)
Unfortunately the dual result which I expected to be true, namely:
(2) If the point of intersection of the diagonals of a circum quad is reflected over its sides, the resulting quad is a cyclic quad
is NOT true, as the perpendicular bisectors of the sides of the resulting quad are NOT concurrent, except in the special cases you've mentioned. These perpendicular bisectors, however, pass through the vertices of the original circum quad - an interesting result in itself.
I've not yet proved these (I'm under loads of final year marking, etc), but do not think they'll be too difficult to prove. Please let me know in the meantime if you or any of your students come up with proofs. Happy exploring!
Regards Michael de Villiers Univ Durban-Westville South Africa