Learn Dutch: Omdat ik de leugen niet geloof, zoek ik de waarheid. - Because I do not believe the lie, I am searching for the truth.

This is the main rule for niet in a subordinate clause! The word omdat (because) pushes the verb (geloof) to the very end of its clause. When that happens, niet moves back with it, sitting directly in front of the verb at the end.

Why doesn't the first part say Omdat ik geloof de leugen niet?

The word omdat introduces a subordinate (dependent) clause. In Dutch, subordinate clauses have a special word order where all the verbs are kicked to the very end of the clause, right before the comma.

We are negating a noun (de leugen). Shouldn't we use geen instead of niet?

We only use geen to negate unspecific nouns, like a lie (een leugen) or just lies (leugens). Because this sentence is about a specific lie (de leugen), we use niet to negate the verb instead.

Why does the second part say zoek ik instead of ik zoek?

This is because of the verb-second rule. The entire first clause (Omdat ik de leugen niet geloof) counts as the first block of information. To keep the main verb in the second position of the whole sentence, zoek has to come right after the comma, swapping places with the subject (ik).