(a) If any township lies partly within and partly without any city, village, or incorporated town and both the city, village, or incorporated town and the township are charged with the duty of providing relief and support to poor and indigent persons in the territory lying within both the township and the city, village, or incorporated town, the county board may upon its own motion detach the part of the township within the city, village, or incorporated town and may annex the detached part to one or more townships wholly within the city, village, or incorporated town.
     (b) The township officers of any township from which territory is detached shall continue as officers of the township until the expiration of the respective terms for which they were elected or appointed and until their successors are elected or appointed and qualified, without regard to whether they reside in the township or the territory detached from the township.
     (c) All property belonging to the township from which territory is detached and all debts of that township before the detachment shall respectively remain the sole property and obligations of that township.