I while trying a hand on coordinate geometry concluded something which appeared amazing to me.There we can make significant use of angle bisector theorem in determining the incentre of a triangle. Now suppose we have vertices given,or anything by which they can be concluded. Now.by using the above stated theorem to determine any point of intersection of any angle bisector through a vertex on opposite side.(by theorem ratio and by section formulka the point. now similarly on that angle bisector too an angle bisector from any of the two remaining sides will intersect and that point of intersection will be our incentre.Thus the required incentre can be obtained by same method as stated above.