Let #O and r# be the center and the radius of the incircle, as shown in the figure.
Given #AB+BC+CA=S#
Let #|ABC|# denote area of #DeltaABC#, #=> |ABC|=1/2*r*(AB+BC+CA)=1/2rS#
given #DE# divides #DeltaABC# into two equal areas, #=> |CDE|=1/2|ABC|=1/4rS ----- Eq(1)# #|CDE|=1/2*r*(CD+CE) ----- Eq(2)#
As #Eq(2)=Eq(1)#, #=> 1/2*r*(CD+CE)=1/4rS# #=> CD+CE=S/2#