Here's a triple homophone for you.
Peak first came into written English in the middle of the sixteenth century. As a noun, the picture above illustrates it well: the highest point of anything, in the literal or figurative sense. The verb is used to describe someone or something reaching its literal or figurative highest point: The quality of printing peaked with the Gutenberg Bible and has been going downhill ever since.
Pique has been a member of English for nearly as long, but still maintains the look of its French origin. As a verb, it means to provoke, often to provoke to irritation: Their bad grammar piqued her to distraction. More commonly, the verb is used in the passive voice: She was piqued to distraction by their bad grammar. It also means to excite, as in the set phrase "to pique one's curiosity." The noun refers to the state of irritation, resentment or wounded pride resulting from having been piqued: The grammar teacher's students sent her into a fit of pique never before seen in the history of the school.
And of course, the last peek has to do with looking at something for a short amount of time. Few seem to have trouble spelling this one, perhaps because of things like peek-a-boo and "sneak peeks."
So as you can see, the three words have little to do with each other, unless someone reaches the peak of his pique as a result of being peeked at.