The illustration shows two circles and three tangents to them. The interesting (amazing?) property is that for any two circles you draw, AB will always be the same length as CD... Would love to hear students talk about how they might prove that this is always the case.
Would they look at special cases? What lines might they draw? Would they try to use a coordinate system and place the circles somehow? And most importantly perhaps, would they persevere if the answer didn't come easily.
It seems like a problem with enough complexity to challenge, but it just LOOKS like it would be true.
I think I would just take it with a scientific approach and see if the theory holds true with most experimental observations. But that's mostly because I'm too lazy to really prove anything. Can't we just take it as a given?
ReplyDeleteCan you be more specific son? What would you try first?
ReplyDeleteWell, without really proving anything, I would try to set it up in geometer's sketch pad and just keep changing things to see if it always holds true.
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