Why is GDP equal to the sum of revenues?
It is right that the whole savings are invested. 

You can use the (simplified) national income and the national output to show the equality. 

The national income is $NI=S+I$

The national product is equal to the national income: $NP=NI$

The national product is equal to the sum of consumption and investment: $NP=C+I$

By using this three equations it can be shown, that $I=S$.

This presumption is made for the IS-Curve at the IS-LM-Model.