Here is a nice exercise in plane geometry. As often in mathematics the statement is quite simple — but the proof is not!

Let \scr{C} be a circle, A,B two distinct points on \scr{C} and M be the midpoint of the chord [AB]. Take two other chords, [PQ] and [SR], that pass through M. Let C (resp. D) be the intersection of [AB] with [PS] (resp. [RQ]).
Prove that M is the midpoint of the chord [CD].

Chords of a circle, midpoint of segments, plane geometry, a theorem in euclidean geometry, butterfly theorem
Suprising! If M is the midpoint of [AB] then also of [CD].