The expressions “by the way” and “incidentally” are called afterthoughts or offhand remarks, and in Spanish there two ways of saying them: a propósito and por cierto.

Examples:
– Por cierto, ¿no estás estudiando para la prueba? (By the way, aren’t you studying for the test?)
– Tienen que entregarme el resumen del libro para el viernes. Por cierto, el libro tiene sólo 40 páginas. (You have to hand in the book summary by Friday. By the way, the book is only 40 pages long.)

A propósito also means “on purpose”.

Mamá, él sabía que yo quería ver la tele y escondió el mando a propósito. (Mom, he knew I wanted to watch TV and hid the remote on purpose.)

A propósito de can also mean “with respect to”, “regarding”.

A propósito de la reunión, tengo que notificarle que ha sido cancelada. (Regarding the meeting, I have to notify you that it has been canceled.)

See you next time!