Se convaincre is a french third group verb. So, this verb is irregular and do not follow a regular conjugation pattern like first or second group verbs. Follow this link to see all the endings of the conjugation of most of the third group verbs :
conjugation rules and endings for the second group verbs.
The verb
convaincre. In grammar, a reflexive verb is, loosely, a verb whose direct object is the same as its subject, for example, "I wash myself". More generally, a reflexive verb has the same semantic agent and patient (typically represented syntactically by the subject and the direct object) are the same. For example, the English verb to perjure is reflexive, since one can only perjure oneself. In a wider sense, the term refers to any verb form whose grammatical object is a reflexive pronoun, regardless of semantics; such verbs are also referred to as pronominal verbs, especially in grammars of the Romance languages. A reflexive verb is always conjugated with a reflexive pronoun. Here is the list of the french reflexive pronouns:
« me »,
« te »,
« se »,
« nous »,
« vous »,
« se ».