Plane Euclidean geometry is good for mind training but research mathematicians rarely have to deal with it in their work. Here is such an occasion: the following problem has naturally appeared in my current work and will probably be included in a paper. It's not hard but I find it amusing.
Let T be a triangulation of a convex polygon P using non-crossing diagonals. So if P has n+3 sides for some n > 0, then T has n diagonals, and we identify T with this n-element set. Consider the following two ways of selecting some special subsets of T.
1. Crossing: take any diagonal d of P, and select the set of diagonals in T crossed by d (here "crossing" means having a common interior point).
2. Corralling: take any four distinct mid-points A,B,C,D of sides of P (say in a clockwise order), and select the set of diagonals in T having one end-point between B and C, and another between A and D.
Claim: these two ways of generating subsets of T are equivalent, that is, if a subset can be obtained by one of them, then it can also be obtained by another.
Solutions are welcome. If nobody finds my solution, I'll post it myself a little later.