FoldPoints
Let a "foldpoint" of a triangle be defined as a place on the triangle to which
the three corners can all be folded simultaneously, such that no folds overlap.
What are the shape and area of the region surrounding all foldpoints for:

an equilateral triangle with side 1?

a 306090 triangle with hypotenuse 1?
If you find any extensions to this problem, let me know. For example:
 Are there any triangles for which no foldpoints can be found?
 For which other convex polygons can foldpoints be found?
 Others?
Source: Original, based on a puzzle in rec.puzzles. Used as Ken's
Puzzle of the Day 4/26/94.
Solution
Mail to Ken