From Problem Solving Strategies by Arthur Engel:
Problem to prove:
There exist three consecutive vertices $A$, $B$, $C$ in every convex $n$-gon with $n \ge 3$, such that the circumcircle of triangle $ABC$ covers the whole $n$-gon.
Proof by the author:
Among the finitely many circles through three vertices of the $n$-gon, there is a maximal circle.
Now we split the problem into $2$ parts:
(a) the maximal circle covers the $n$-gon.
(b) the maximal circle passes through three consecutive vertices.
We prove (a) indirectly. Suppose the point A' lies outside the maximal circle about triangle ABC where A, B, C are denoted such that A, B, C, A' are vertices of a convex quadrilateral. Then the circumcircle of triangle A'BC has a larger radius then that of triangle ABC. Contradiction.
We also prove (b) indirectly. Let A, B, C be vertices on the maximal circle, and let A' lie between B and C and not on the maximal circle. Because of (a), it lies inside that circle, but then the circle about triangle A'BC is larger than the maximal circumcircle. Contradiction.
Here is what I don't understand about the solution:
Is there sufficient proof for why the circumcircle of $A'BC$ has a larger radius than that of $ABC$? (for both the cases (a) and (b)). Also, how is assuming the quadrilateral to be convex any helpful?
If I draw it out, I can kind of see that $A'BC$ has a larger radius than that of $ABC$, but it doesn't make sense intuitively how having points outside (part (a)) and inside (part (b)) of the circumcircle would both lead to larger radius circles?
Any help/suggestions would be much appreciated.
I think it is a good move to consider the largest circle incident with three of the given points, and then to prove (a) and (b) in turn.
But the verbal arguments put forward by the author as an "indirect proof" of (a) and (b) can only serve as guidelines, and not as an actual proof. They would have to be substantiated with figures. It then would become obvious that several cases have to be considered; in particular whether the primary triangle $ABC$ contains the midpoint of the maximal circle in its interior or not.