There was a real Old English word Þrīhyrned meaning 'three-cornered' or 'triangular', but this original meaning does not seem to apply directly to Thrihyrne in the White Mountains. We're told that the mountain had three separate peaks (actually described as 'three jagged horns' in The Two Towers III 7, Helm's Deep) so in this particular context the name was apparently intended to literally mean 'three-horned'.
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