Star-Triangle transform

Also known as the Tee-to-pi or Y-Delta transform, the star-triangle transform allow to transform a star shaped tripolar electrical network to a triangle shaped network with he same behavior and backwards. Suppose we start from the following star/T/Y (the two bottom nodes are in fact only one same node, as they are supraconductively connected) :

     ------   ------
0----| A  |---|  B |----0
     ------ | ------
            |
            -
           | |
           |C|
           | |
            -
            |
0-----------------------0

The triangle/Pi/Delta we want to transform to is the following :

          ------   
0---------| c  |--------0
     |    ------    |
     |              |
     -              -
    | |            | |
    |b|            |a|
    | |            | |
     -              -
     |              |
0-----------------------0

The transformation formulas are the as follows :

a=(A.B+B.C+A.C)/A
b=(A.B+B.C+A.C)/B
c=(A.B+B.C+A.C)/C

And backwards :

A=(b.c)/(a+b+c)
B=(a.c)/(a+b+c)
C=(a.b)/(a+b+c)

This is esily proved by calculating the impedance between each node. This gives a set of three equations that can be added / substracted to isolate any of the variables.