You are hiking from point A to point B. Suppose you leave point A at 9 AM on Monday and after an arduous trek where you walk at your own pace and take as many breaks as you want, you arrive at B by sundown. You spend the night at B and on the next morning (Tuesday) you leave at 9 AM to head back to A. On your trip back, you also walk at your own pace and take as many breaks as you want and you arrive back at A by sundown. Importantly, the path you take to go from A to B is the same as the path you take to go from B to A. Show that there is a point C in this path where you were present both on Monday and Tuesday at the same time of the day. For example, if you walked from A to B on Monday at a constant speed v (without stopping) and arrived at B at 3 PM (on Monday) and you took the same path back on Tuesday walking at the same constant speed v (also without stopping), then on both Monday and Tuesday you would have been at the mid point of the journey at exactly 12 PM. Required:
Your task is to show that such a point exists no matter how you hiked from A to B and back from B to A as long as you took the same paths and left at the same time of the day on both days (9 AM in this example)