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A propósito vs. adrede vs. intencionalmente

The words a propósito, adrede, and intencionalmente in Spanish all translate to intentionally or on purpose in English. However, they are used in different contexts and have slightly different connotations.

A propósito

B1
A propósito can mean intentionally, but it is often used to mean by the way or incidentally. It can also refer to something being relevant or pertinent.
A propósito, ¿has visto a Juan hoy?
(By the way, have you seen Juan today?)
Hiciste eso a propósito para molestarme.
(You did that on purpose to annoy me.)
Tu comentario es a propósito para la discusión.
(Your comment is pertinent to the discussion.)

Adrede

B1
Adrede strictly means doing something intentionally or on purpose, usually with a negative nuance. It often implies that the act was committed with malice or harmful intent.
Rompiste mi vaso adrede.
(You broke my glass on purpose.)
No lo hizo adrede, fue un accidente.
(He didn't do it on purpose, it was an accident.)

Intencionalmente

B2
Intencionalmente translates directly as intentionally. It is often used in formal contexts or when referring to deliberate actions or thought processes.
Fue despedido por actuar intencionalmente contra las políticas de la empresa.
(He was fired for intentionally acting against the company's policies.)
La jugadora intencionalmente cometió una falta para detener el juego.
(The player intentionally committed a foul to stop the game.)

Summary

A propósito, adrede, and intencionalmente can all be used to convey the idea of doing something intentionally or on purpose in Spanish. However, a propósito is more versatile and can also be used to express pertinence or to transition into another topic (by the way). Adrede typically implies intentionality with a negative connotation. Lastly, intencionalmente is more formal and often refers to deliberate thought processes.