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"pointed top," 1520s, variant of pike (n.4) "sharp point." Meaning "top of a mountain" first recorded 1630s, though pike was used in this sense c.1400. Figurative sense is 1784. Meaning "point formed by hair on the forehead" is from 1833. According to OED, The Peak in Derbyshire is older than the word for "mountaintop;" e.g. Old English Peaclond, for the district, Pecsaetan, for the people who settled there, Peaces ærs for Peak Cavern; sometimes said to be a reference to an elf-denizen Peac "Puck."
1570s, "to rise in a peak," from peak (n.). Figurative meaning "reach highest point" first recorded 1958. Related: peaked; peaking.