Why does the current in autotransformer motor starting gets reduced squared in relative to the voltage?
The current in the motor scales linearly with voltage at the motor. However, at the transformer input the current is scaled down by the transformer ratio a second time.

So if the transformer ratio is X, the motor voltage is VLINE &times; X. The motor current is INOMINAL &times; X, and the power is VLINE &times; INOMINAL &times; X2.

To get the value of the line current at the input of the transformer, divide the power by the line voltage, and you're left with INOMINAL &times; X2.