I came across this remarkable result in Futility Closet:

“On each of these two black lines is a trio of red points marked by the same distances. The midpoints of segments drawn between corresponding points are collinear.

(Discovered by Danish mathematician Johannes Hjelmslev.)”

This result seems amazing and mysterious. I wondered if I could think of a proof. I found a simple approach that did not use plane geometry. And suddenly, like a magic trick exposed, the result seemed obvious.