Fermat posed a challenge problem thus: Given three points find a fourth in such a way that the sum of its distances from the three given points is a minimum. The solution point is called Fermat Point (FP). The problem involved three given points and the minimization of sum of three distances. The solution contained some interesting special cases which involved the three given points but only two distances whose sum was a minimum. We found the special cases provide a simple method for exposing the inconsistency between FP and Fermat’s least time principle (FLTP). The perfect setting for our finding was provided by the natural phenomena of reflection and refraction of light. In the application of FLTP to these processes also, we have the same conditions of three given points and two distances. The three points are: the end points of the broken line path and the point of incidence. The two distances are: the lengths of the two broken line segments - travelled before and after reflection or refraction. We show in this article that FP and FLTP lead to contradictory results about the point connecting the given points that provides the minimal sum of the distances. In optimization parlance this means that FP and FLTP give different points to locate a service facility catering to three given towns. Our result leads to the conclusion that FP and FLTP are mutually inconsistent. Simply put, we pitch FP against FLTP and show the inconsistency between the two.