Is the Current in the Middle of a Balanced Wheatstone Bridge Always Zero?

[mafra]

Homework Statement

Show that the current in the middle is zero if all the resistors are equal

The Attempt at a Solution

KVL for left and right:
V - R*i1 - R*i2 = 0
V - R*i3 - R*i4 = 0

i1 + i2 = i3 + i4

KCL:
i1 + i3 = i2 + i4
i1 + iG = i2
i3 = iG + i4

I must prove that i1 = i2 or i3 = i4 but all I get is i1 = i4 and i2 = i3 combining KVL and KCL

 

[azizlwl]

For the current in the middle to be zero, points A and B are at the same potential.