My question is regarding Steiner's (incomplete) proof for the Isoperimetric problem, as presented in the book What is Mathematics. In a critical step, Steiner asks to readjust half the 'assumed' maximum area in such a way that the two end lengths become 90 degrees at the curve.
Then he argues that this has a greater area (because for two sides, the right angle produces maximum area) hence there is a contradiction—we found another similar curve with a greater area.
He proceeds to say that it must be a right angle at all points. Thus it's a circle.
I am struggling to accept the 'readjustment' argument. I just can't see it. Two questions:
- How can we readjust the sides without effecting the shaded area?
- How can we readjust the sides without effecting the total length (which is a constant i.e. $L/2$)
In an article by Viktor Blåsjö I read that we have to assume the areas 'glued' on top of the triangle, but then wouldn't re-adjusting the areas create a kind of a kink at the point?
Imagine the shaded regions are made of cardboard and the point $O$ is a hinge. Opening/closing the hinge won't affect the area or outer perimeter of the shaded cardboard pieces.