If we have one random triangle and bisected with respect to the longest edge, i.e 
Left the initial triangle with the longest edge is highlighted by double line. If we bisected this triangle, then we get the right. My question is: the longest edge of the two new triangles are the edges of picture with double line? Thanks!
No.
You can always construct a triangle with the biggest side c at least 2 times greater than one of the others two. Consider a triangle with sides 4, 10, 7.