Geometry is one of the oldest branches of mathematics, and many famous problems have been proposed and solved in its long history.
What I would like to know is: What is the oldest open problem in geometry?
Also (soft questions): Why is it so hard? Which existing tools may be helpful to handle it? If twenty great geometers of today gathered to work together in the problem, would they (probably) be able to solve it?
P.S. The problem can be of any area of geometry (discrete, differential, etc…)
Does every triangular billiard have a periodic orbit?
For acute triangles, the question has been answered affirmatively by Fagnano in 1775: one can simply take the length 3 orbit joining the basepoints of the heights of the triangle.
For (generic) obtuse triangles, the answer is not known in spite of very considerable efforts of many mathematicians. Apparently, A. Katok has offered a 10.000$ prize for a solution of this problem.