Learn French: Il a intelligemment construit cet argument brillant. - He intelligently constructed this brilliant argument.

It comes from the masculine adjective intelligent, which ends in -ent. To form the adverb, you replace the final -nt with -mment, giving us intelligemment. Even though it is spelled with an E, the ending sounds exactly like -amment.

Why is intelligemment placed between a and construit?

In the past tense (passé composé), short adverbs and common adverbs of manner usually go between the helping verb (here, a) and the past participle (construit).

Why does the sentence use cet instead of ce for the word "this"?

Argument is a masculine singular noun, so we would normally use ce. However, because argument starts with a vowel, ce changes to cet to make it flow better when spoken.

Since we are learning to make adverbs out of words ending in -ant, why is brillant left as is?

Because brillant is acting as an adjective here, describing the noun argument (a brilliant argument). You only change it to brillamment if you need an adverb to describe how an action was done (for example, "he succeeded brilliantly").