In Spanish class I learned that the right way to say "by the way" in Spanish was a propósito, but I've recently seen it translated por cierto. How exactly are these two Spanish phrases used, and are they universal through the Spanish-speaking world? Which is a more accurate way of translating "by the way" (to introduce different but related new topic of conversation)?
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"Por cierto" and "a propósito" are synonymous. Both can be correctly translated as "by the way." Neither are too formal nor too slang.
As a native speaker I can say both terms are widely accepted.
Both are widely accepted, but you could also use this other:
-Ya que estamos... (Now that we are talking about this..)
And then include a new question/comment/information/whatever related to the subject of the conversation.
They're basically synonymous but have SLIGHTLY different meanings. "A propósito" seems a bit closer to "by the way." "Por cierto" is more like "certainly," or "surely," and is a bit more emphatic. I might "translate" it as "BY THE WAY" [emphasis added].
NOTE: "Por cierto" is NOT commonly used as "certainly" in Spain, and we are still trying to determine whether or not it is used this way in parts of the "New World." But the as far as we can determine, the "authorities" permit this usage. "True" meaning of "por cierto"