What I refuse to do is to always justify every topic and problem that I teach as being useful to every student for every day of their future lives. I won't even guarantee that every topic will be "useful" even just once for each student. I can never do that because I know that every kid is different and will have different needs. Some kids are losers who will grow out of being losers -- what you were sure was useless last year may suddenly become very useful tomorrow.
The problem with always requiring "a real-world application or you'll dump the material" is that all of math can be reduced to this absurd level if you try hard enough. As can poetry. And chemistry. and history. and art. (and grammar and writing, if you took my principal's example). Every topic can be eliminated by somebody.
"When am I EVER going to use this?" becomes a weapon instead of a question.
Why not teach it "just because?"
Why should my short-sighted, intensely hormonal, spring-addled students have the right of refusal over anything they don't immediately see a purpose for? I can see a couple of blue-collar uses for those circle and tangent rules above. If I can't convince a student that a machinist, draftsman or custom motorcycle builder would need at least an understanding of this stuff, should it be eliminated? I don't think so.
Sometimes they just need to follow the leader. "We're doing it because I think it's neat, it's part of the course, it fits here and we have the time."
As math teachers, we need to refocus the question and answer it ourselves. We need to take the question out of the mouths of the lazy and decide what does, indeed, make for a good curriculum. If something like those circle theorems can be used in any meaningful way either later in life or later in the week, then WE should decide. If they have no purpose other than intellectual curiosity, then they have that going for them, don't they?
Geometrical theorems are rarely useful in the raw as it were, but in combination with other knowledge, make a different problem solvable. It's a polygon inscribed in a circle -- or is it a bolt-pattern for a truck wheel? Tangents and circles, central and inscribed angles, external angles come together all the time in machining and manufacture.
To answer the original question:
Every time I see these theorems, I think of the guys on Junkyard wars who recited geometrical theory when building a go-kart. They won.
Some other comments:
"My take on it is that the people behind the unified 10-12 curriculum looked at circle geometry and asked these same questions of "why?", had no answer, and got rid of them. " The commenter didn't get rid of them but the powers-that-be did.
"I don't consider "it's interesting" a good enough justification, because there are plenty of things that are both interesting and relevant to what students will later see of pure mathematics." In other words, justify or get replaced.
"it's a mathematical dead end." and "but I can't even think of a later pure-mathematics connection. Shouldn't we be suspicious of anything that's a "stub" in the curriculum?" I can't think of a reason for it, so let's be suspicious of it?
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I was thinking about this the other day, and it seems to me that anything you learn thoroughly becomes useful eventually. If you learn it, it will turn out the be useful--math, poetry, french, whatever. All this stuff that you didn't think you'd ever use helps you make a connection, or add something to a (work) project, or catch an error before it embarrasses you or your boss or loses the company money. If you don't learn circle properties, you won't ever notice that you didn't, but if you learn them well, You'll find out that it helps you design a web animation/design windows/build a go-cart. I've found uses for enough unlikely bits of knowledge over the years, that I've decided that almost everything is useful--at least for me.
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