Here's another proof of the math being used. Please try ang get yourself a copy of the book by John J. Winders, Jr. of PPL Electric Utilities entitled "Power Transformers and Their Applications". LINK
On page 96:
"Define the following ratios of impedances, w and x, as follows. (Note: In general, these ratios are complex numbers because the impedances are complex numbers.)
On page 97:
"The voltage drop around the closed Δ path must equal zero:
On page 96:
"Define the following ratios of impedances, w and x, as follows. (Note: In general, these ratios are complex numbers because the impedances are complex numbers.)
w = Zb/Za → Zb = wZa (3.10.1)
x = Zc/Za → Zc = xZa (3.10.2)"
x = Zc/Za → Zc = xZa (3.10.2)"
On page 97:
"The voltage drop around the closed Δ path must equal zero:
iaZa + ibZb + icZc = iaZa + ibwZa + icxZa = 0 (3.10.6)
ia + w ib + x ic = 0 (3.10.7)"
ia + w ib + x ic = 0 (3.10.7)"