TITLE: A question about axes of symmetry in the plane.
QUESTION [5 upvotes]: Suppose J is a Jordan curve in the Euclidean plane E and that there are two perpendicular straight lines 
in E, each of which is an axis of symmetry of J. Does the intersection point of these two lines always
necessarily lie in the interior of J?

REPLY [7 votes]: Here is a more general fact: if $J$ is central-symmetric w.r.t. a point $O$ (in our case, this is the intersection of the axes), then $O$ is inside.
Indeed, let $D$ be the domain bounded by $J$. We know it's a topological disc. And it's central symmetric w.r.t. $O$. Now if $O$ is outside, then the symmetry is a map from $D$ to itself without fixed points, contrary to Brouwer fixed point theorem.
If $O$ is on the boundary (i.e. on $J$), the the central symmetry is an involution of $J$ with exactly one fixed point. But there are no such involutions of the circle.