Let $\gamma$ be a continuous Jordan curve comprised of three smooth pieces. Suppose that at the points that each two of the pieces meet, the tangent lines to the two pieces are different. Does there always exist a circle inside $\gamma$ that intersects each of the three pieces at exactly one point?
The answer is most likely no. Then what are some sufficient conditions on $\gamma$ that would guarantee such a circle exists?
Are you sure the answer is most likely no? I would say, simply by thinking with Bolzano's theorem you could always find such a circle... I can't seem to visualize how pathological the curve could be in order to not being able to find a solution... excuse me if I'm being too rough here