[ Polygons (inequalities)

Each side of a convex hexagon is greater than 1. Is there always a diagonal of length greater than 2 in it?

#
Construct triangles on the sides of an equilateral triangle with side 2, outward.

## Solution

Take an equilateral triangle with side 2. From the midpoints of its sides, erect perpendiculars outward, mark off segments of very small length (say, length 0.01) on them, and connect the resulting points to the vertices of the triangle. It is easy to see that the sides of the resulting convex hexagon are greater than 1, but all its diagonals are no more than 2.

## Answer

not always.