To make mad or drive to madness; to cause someone to become angry or insane (archaic).
From 'be-' (prefix intensifying or completing the action) plus 'mad' (adjective/verb). Part of the productive 'be-' verb system in Middle and Early Modern English.
The 'be-' prefix could turn almost any adjective into a verb in Middle English—so 'bemad' is just one of hundreds of similar constructions like 'besmirch,' 'bedazzle,' and 'befriend' that show how creative English was.
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