Howcome it is safe to assume the the voltages are equal in this picture?
howcome it is safe to assume the voltages are equal

By symmetry.  Since all resistors are equal and the net above the center is a mirror image of the net below the center, both those nets will have equal effect on the center voltage.  Recognizing this symmetry allows you to see immediately from inspection that all three center nodes are at the average of A and B.  In this case, since B is defined as 0, the center nodes are also at A/2.

Or, you could do it the hard way.  First, solve for the voltage at C.  Do this by combining series and parallel combinations of resistors from the right end, converting to Thevenin equivalents, etc.  Eventually you will reduce the net to two equal resistors forming a divider from A to ground.  Obviously C is then A/2.  With that knowledge, you can find the Thevenin equivalent what is driving D, etc.