1. It's interesting. Circumcenters are equidistant from original vertices A, B, C, so they give rise to certain problems such as "where is the best location for placing a new hospital/communication tower?"
2. The geometry of circumcenters is beautiful; it can be found by folding each point on top of each other point* (thereby creating 3 perpendicular bisectors kinesthetically), and afterwards its location can be verified by drawing a circumscribed circle that goes through all original points. This also helps to reinforce the circular property of equidistance.
*And this year, since my kids were the ones that came up with the "folding" algorithm, it makes total sense to them why that line is the locus of equidistant points from vertices, and why the circumcenter must lie on the point of concurrency of all those perpendicular bisectors.
3. The algebra of perpendicular bisectors is a nice way to loop back to line equations, midpoints, and perpendicular slopes. For the more advanced geometry kids, the algebra of finding a non-integer coordinate circumcenter also brings back systems of equations algebra, since that is the point (x, y) that needs to satisfy all perpendicular bisector equations.
But, last year my students had a lot of trouble with the algebra part. This year, we're done with the kinesthetic part and they don't seem to have any trouble with the overall concept. We started looking at the algebra, and originally I started it the same way I did last year -- by running them through the list of properties that a perpendicular bisector should have, and using those properties to help us write the equation -- which sounds good in theory. These regular Geometry kids can follow conceptually what I'm saying (since they have a strong conceptual understanding of circumcenters through our various activities and demos), but then when I let them follow up on the algebra example by doing one of their own, all hell broke loose.
Naturally, I thought in my head: I need to back the heck up!!
So, I made the following worksheet for them yesterday (ouch, algebra on a Friday!), and it went really well. The worksheet had different problems of varying difficulty; initially I would give them either the original slope or the midpoint already found (or both), and expect them to find the missing pieces and to find / graph the perpendicular bisector equation. (They graphed to check visually whether their perpendicular bisector equation "looked right" relative to the original segment; I refused to tell them whether or not their equations were correct on the first page.) Then, the worksheet scaffold up to them doing the whole process of perpendicular bisectors by themselves, and finally to finding the circumcenters using the intersections of those graphed perpendicular bisectors.
I found that by breaking it up like this into little pieces, it finally started to make sense to kids and they were able to "see" how the perpendicular bisectors, once they had finally understood how to find them, would lead them graphically to the circumcenters. (To be sure, these regular kids were having a lot of basic algebra issues. This exercise also helps them to zoom in on those, because it makes it relatively easy to figure out which part your mistake must be coming from, if half of the problem was already done for you. Because I'm trying to be less helpful and to force their independence, I also told them they needed to find their own algebra mistakes and not rely on me. Only very seldom would I help a kid diagnose that their midpoint wasn't correct, for example, by asking the kid to find that point on the graph and telling me whether that location appeared to be the correct midpoint.)
--Score! ...You know, it's funny. These are the traps I should have been able to avoid even as a first-time Geometry teacher last year. You can't teach kids a whole algebra process at once, even IF they already have the conceptual foundation. They need to already have a solid understanding of the individual algebra pieces, before they can start to put the whole process together end-to-end. It's something that I always forget the first time I try to teach something. But for some reason, it never occurred to me last year that this was what I was doing wrong. More practice doesn't automatically lead to more understanding; better and more thoughtful practice leads to more understanding!!