Discussion: A good example of the extremal principle and proof by contradiction. The crux move is to order the lengths of the line segments by ascending size and assume that no triangle can be made from three. Then the sum of any two consecutive lengths must be smaller then the succeeding one but this gives rise to a contradiction. In a very nice way the Fibonacci sequence arises.

Written response: Well done. Good idea to order the lengths and get a contradiction under the assumption that no triangle exists.