In the first case, when you say “la barrière de Corail”, de stands for a particle indicating the substance that makes up the thing you’re talking about. It means that the reef is composed of coral, you could implicitly say “la barrière (faite de) corail”. Normally when talking about the matter an object is made of, people use en (e.g. un gilet en laine: a wool jacket ; un mur en béton: a concrete wall). But here if you say “la barrière en corail” it’ll sound like coral has been crushed and turned into bricks in order to make a wall. “La barrière de corail” rather sounds like coral has been growing all around and has spontaneously formed a reef. It also sounds more elegant. (“La barrière du corail” would sound like you’re talking about a specific coral as a topic).

About the “Tropique du Capricorne”, the le from du (= de le, just to be clear) simply comes from the fact that capricorn is an astrological sign, which are usually referred to using an article: La constellation du Capricorne; Être né sous le Capricorne. “Tropique de Capricorne” would sound like Capricorne is a person, like a great explorer etc… (just like in “Détroit de Magellan”). That’s it, hope it helps!

il y a plusieurs interprétations pour ces termes, et pour éviter la confusion on prend ces exemples :

• Devoir de français.

• Devoir du Français.

• Devoir de la police française.

-Dans la première phrase on parle de la langue et donc d’une caractéristique.

-Dans la seconde on parle de la personne (masculine).

-Dans la dernière on parle de la personne (féminine).

Remarque : il y a un autre usage pour le (Du/De/De La), qui sont des articles partitifs, exemple, "Il mange du pain" !