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Why do we say "Je suis obligé DE quitter" and at the same time "Il m'oblige A quitter"?
What is the rule about these prepositions? Is there anything special about prepositions with participles?
Yes, the rule is être (adjectif / participe passé) de (infinitif) with very few exceptions (e.g., être prêt à).
Verbs like obliger, on the other hand, don't have a nice, neat rule like that. You have to learn which verbs take à, which verbs take de, etc.
Required, but never shown