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Solution
This can easily be shown in an algebraic way. But for those not conversant with algebra I have used a geometrical way to prove it instead. This geometrical way represents the exact same reasoning as one would do with algebra.

These barrels contain one glass more each of the red and the white wine than what is needed. The barrels are shaped so that you get perfect squares of red and white wine.

Throw away that extra glass from each barrel. We now have the volumes we wanted, and we don't have squares anylonger.

Take one glass of red wine...

...and put it into the barrel with white wine. Note that the wine in the right barrel now makes a perfect square, i.e. its height is the same as its width, and of course also the same as the width of the left barrel.

Take one glas of mixed white and red wine (the vertical portion)...

...and put it into the red wine. The red part in the white is now of the same size as the white part in the red.