Is the fast of one who intended to fast at night and woke up with the dawnathan while on the fifth day of her menstrual period, knowing that she usually does not purify until after four days, invalid?

As long as you were not certain of purity before the dawn broke, your fast is not valid, and you must make up that day. This is because the default state is the continuation of menstruation, and because an incident is estimated by the nearest time. Purity is an incident, so its occurrence is estimated to be after the dawn broke, and therefore the fast is not valid at that time.