The Butterfly Theorem
Image: Wikimedia Commons

Draw a circle, choose any chord PQ, and draw two further chords AB and CD through its midpoint M. Now, if AD and BC intersect PQ at X and Y, M will always be the midpoint of XY.

In Icons of Mathematics (2011), Claudi Alsina and Roger Nelsen write, “The surprise is the unexpected symmetry arising from an almost random construction.” The theorem first appeared in 1815.