In a question, we are asked to calculate the equivalent resistance between the points A and B in the following circuit. 

![enter image description here](https://i.stack.imgur.com/YptKH.jpg)

For that the first step involved in the solution is that the potential at points $ C$, $C\_1$ & $C\_2$ are equal: this is what I don't get. 

Suppose the current is flowing from point A; then equal current flows through all the three branches and when it passes through the resistance *r*, the potential drop across these resistances are equal and therefore, the potential at the points C and D should be the same, shouldn't it?

Then, why is it not so? Why is the potential of $C$ and $C\_1$ same instead (even though the current has to pass through an "extra" resistance $\frac r2$in one of the branches)?

I know I am wrong because I'm getting absurd results from my reasoning.

EDIT:![enter image description here](https://i.stack.imgur.com/IN7y3.jpg) In this picture all the resistances are of the same value, why does equal current flow through the three branches originating from A? 
How did we decide here whether the points B, B' and D have the same potential?

Contrasting it with this image (again all the resistances are the same):
![enter image description here](https://i.stack.imgur.com/xxzpb.jpg)

Why don't the currents passing through the three branches originating from A have the same value in this case?
 (The only difference in this circuit is that the points A and B lie on the same edge of the "cube".)