Necessarily a Circle? 2

Let's place the centre of the shape at $O(0,0)$ and let a point on the edge of the shape be at $A(1,0)$ . Since the shape must be the same on reflection from any angle, we can take point $A$ and reflect it around every possible line through the centre. The curve created by all these points will be the shape we are looking for. An example of one possible reflection is shown below:

Notice that since $\triangle AOM$ is a reflection of $\triangle BOM$ , $OA=OB=1$ , and so the locus of the point $B$ is simply one where it keeps a distance of 1 unit from the origin, which is the definition of a circle.