Hexagon: Diagonals and sides (If and only if?)

[BMAD]

 

Each of the side of the convex hexagon (6-angle) is longer than 1. Does it necessary have a diagonal longer than 2?

 

 

 

Each of the main diagonals of the convex hexagon is longer than 2. Does it necessary have a side longer than 1?

[k-man]

1) No. Here is a counterexample. Start with an equilateral triangle with a side length of 2 (or even slightly less than 2). The vertices of the triangle will become 3 of the vertices of the hexagon. The other 3 vertices will be found by extending the midpoints of the triangle's sides outward just enough, so that each new side formed is longer than 1, but all the diagonals remain 2 or shorter. 

 

2) Yes. The easiest way to see it is to reverse the necessary condition into a sufficient condition - "A convex hexagon with all sides equal to (or less than) 1 cannot have all main diagonals longer than 2". It's kind of obvious to me, so I'm not sure what exactly needs proving. All main diagonals are the same when the hexagon is regular and are equal 2. Just imagine a hexagon made out of rigid sticks with swivel connections - making one diagonal longer forces other diagonals to be shorter, etc. It's impossible to stretch all main diagonals without stretching at least one side of the hexagon.

[gavinksong]

I want to say 1) yes and 2) no - proof forthcoming.

 

For (1), my first thought is that, assuming that all the sides are 1+, making one diagonal shorter makes another longer, so the hexagon with the shortest length that none of its diagonals exceed I imagine would be a regular hexagon. On a regular hexagon, if all the sides are 1+, then all the diagonals must be 2+. I will try to come up with more rigorous proof.

 

However, for (2), it's easy to come up with a counterexample. One way is to first draw a line shorter than 1, then draw draw crossing diagonals longer than 2 from each endpoint. The endpoints of these two diagonals are vertices number 1, 2, 4, and 5 of a counterexample hexagon.

[BMAD]

It appears that we have disagreement, both of you have provided proofs where the other has provided a counter example

[gavinksong]

I think I'm the one who is at fault.