No polygon has the same area as the difference between its inscribed and circumscribed circles. The inscribed circles must touch every side and the circumscribed circle must touch each vertice. I have proved this for some simple cases but failed to prove it generally. Or is there any counter-proof? Please help.
dbx proved that it does not hold for some irregular polygons. A big round of applause for him on cracking that tough nut? So some new questions to ponder about:
Are there a finite number of irregular polygons who disobey this hypothesis?
Are there a finite number of irregular polygons who obey this hypothesis?
Could anyone give any more examples of such polygons who do not obey this hypothesis.
Also thanks to Ross and anderstood who proved this does hold for all regular polygons.
I have expanded on this idea: There is no such polygon whose perimeter is equal to the difference between the circumferences of its circumscribed and inscribed circle .
I may also continue this onto the third dimension if I get conclusive results for the above post. All the best!