Question 663868
Notice how AP = AT + TP and AP = AB + BP


Since AB = TP (given), this means 


AP = AT + TP and AP = AB + BP


becomes


AP = AT + TP and AP = TP + BP


AP = AT + TP and AP = BP + TP


So this means that


AP = AT + TP


BP + TP = AT + TP


BP + TP - TP = AT + TP - TP


BP = AT


AT = BP


which proves that segment AT is congruent to segment BP