Partially answered in comments:
Don't dichotomize a continuous variable. In addition, looking for a data manipulation to minimize your p-values is data dredging. – gung
See this page and many others on this site why you should not try to break up your continuous predictor this way, for this or for any type of regression/classification scheme. Much better to learn about the actual relation between your continuous predictor and outcome when standard clinical variables are taken into account in your Cox model. – EdM
To which OP responded: In my case, the cut-point is needed to find the proper criteria to make a diagnosis of a particular disease. Answer to that is: If the goal is to find a decision cut-point to make diagnosis for a particular disease, then do that after model construction and estimation, estimation and decision is different problems. In particular, decision tresholds could vary with patient caracteristics. See for instance the F Harrell quote in Calculating F-Score, which is the "positive" class, the majority or minority class?