János Kurdics published this interesting problem in The Math Connection Linkedin group:
Given a triangle, find the shortest line segment that divides the triangle into two regions of equal area. A solution that does not involve calculus is preferred.
The solution that I know is very nice, and gives quite a workout in elementary trigonometry.
János says that he is working on three-dimensional analog (plane that divides a tetrahedron into two regions of equal volume and gives smallest possible cross-sectional area with the tetrahedron) but has only solved it for a regular tetrahedron.