I came across this at Futility Closet, a blog of such diverse interest and appeal that you just have to go there to see for yourself.
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.