Faster than a short sentence, more powerful than a rebuttal, and able to refute strong statements in a single syllable... DOCH

The multi-functional word doch, in some cases fulfilling the role of “but” (or “still” or “nevertheless”) in English, has the capability to do with one word what in English requires an entire phrase: to negate a preceding negative statement with an affirmative. In German, English phrases such as “On the contrary” and “Yes, I do”  can be replaced with the monosyllabic doch.

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Before you attempt to prematurely launch a speedy doch at an angry policeman or boss, however, let us first examine the simpler usages of doch translated as “but”:

Doch jeder weiß hier, das ist die Luft, die brennt

But everyone here knows, that's the air that's burning

Caption 45, 2raumwohnung - 36 Grad

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And “however”:

Doch am Ende dieses Weges wird Europa stärker aus der Krise hervorgehen.

However, at the end of this path Europe will go forth from the crisis stronger.

Caption 38, Angela Merkel - Neujahrsansprache

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And “indeed”:

Doch das scheint sich nun geändert zu haben.

Indeed, now this seems to have changed.

Caption 7, Apple-Trojaner - Wie man ihn beseitigt

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And as an affirmative:

Aber wir hatten 'nen guten Start in Braunschweig

But we had a good start in Braunschweig

und machen eigentlich ganz gut weiter, doch.

and have actually continued quite well, really.

Caption 8, Cassandra Steen - Interview

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Here are some examples showing the full power of doch as a negation:

Der Eierkumpel von nebenan, der wusste nämlich nicht, was Pi ist!

The egg pal [egg seller] next to you, he didn't know what pi is!

-Doch!

-Yes, he did!

Captions 22-23, Welt-Pi-Tag - Unser Leben mit der Kreiszahl

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Ich weiß nicht, ob man als Frankfurter mal nach Mainz fährt.

I don't know if you'd go to Mainz as someone from Frankfurt.

-Doch, war ich auch schon...

-Yes, you would. I've been there too...

Captions 19-20, Museumsuferfest - Jazzmusiker Daniel Stelter

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Here we see doch first in the affirmative sense, then in the more sophisticated negation sense, all in a single caption: 

Männer kommen doch nicht hierher, oder? -Doch, natürlich.

Men don't really come here, or? -Yes they do, of course.

Caption 24, Waxhouse - Brasilianische Haarentfernung

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So remember the the two main uses of doch: 

1. As a simple affirmative (“really”), negating adverb (“however”), or transitional word (“but”):

Er hat es doch nicht getan.

He did not really do it.

Doch wissen wir, was es bedeutet.

But we know what it means.

2. As an all-powerful negation to a previous negative statement: 

Er hat es nicht getan. -Doch.

He did not do it. -Yes, he did do it.

Sie waren nicht dort. -Doch.

They were not there. -Yes, they were there.

By learning the use of the powerful doch, you will be able to negate, with a minimum of syllables, any negative statements with which you disagree!

Hast du nicht verstanden? -Doch!

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Learning suggestions:

Get some great explanations and examples of doch here and (as a PDF) here. 

More advanced learners will enjoy this explanation in German, from Wiktionary.