broach, breach and breech
It all starts with the Latin brocca meaning ‘a spike or pointed instrument’. In Middle English the word broach, picked up from the French broche, referred to a variety of tapering bits of wood or iron, so it could mean ‘a skewer, a sturdy pin, a lance or spear’. Finally it came to mean the decorative pin that you put in your clothing and, in that meaning, transitions quite early form the French broche into brooch.
From a noun it moves to a verb with the sense, understandably, of ‘to stab or spike or skewer’. In particular it refers to making a hole in a cask of wine in order to tap the contents. That sense is used figuratively in relation to a number of things which are opened up in order to extract something, and finally it is used in relation to opening up a topic for discussion.
A breach is a hole or gap created in something and comes from Old English where it is perhaps related to the Germanic word break. As a noun in the literal sense of ‘a break or fracture’ it became obsolete in the 1600s but it had and still has a very productive life used figuratively, as in a breach of contract. It still has a literal meaning when used as a verb, so it is still possible to breach a physical obstacle of some sort. And it is from this sense that we get the whales breaching. They break through the surface of the water.
A breech is an obsolete word for the item of clothing that covers your backside. It survives in the plural form breeches. To breech someone is again obsolete and means ‘to put breeches on them’.