I'm trying to solve this question in the classical Do Carmo's differential geometry book (page 23):
Assume that all normals of a parametrized curve pass through a fixed point. Prove that the trace of the curve is contained in a circle.
My counterexample is parts of circles with different radii:
Am I right? If not, I need a hint to solve this question.
Remark: Please don't give me solutions, I already have them.