Quandaries
and Queries 

Is it possible to construct a triangle with sides that are three consecutive Fibonacci numbers? Could you answer with algebra. Thanks Marcelle 

Hi Marcelle, The answer of course depends on what you mean by a 'triangle'. This is a debate I have with students in my classes (sometimes classes of high school teachers). Fibonacci requires a_{n2} + a_{n1} = a_{n} So the third (longest) side is the sum of the other two sides. This will require the three points to be along the same line! Or, in algebra terms we note in a triangle that a+b ≥ c and some people would actually put this as a pure inequality a+b > c. I have not problem allowing three points on a line to create a 'triangle'.
However, if your group (or your teacher) has agreed that you will NOT use the word 'triangle' except when the three points are not on a line, then you have the conclusion you want. By the way, if you wonder about triangles with three points on a line, you might wonder about 'trilaterals' (something formed by three intersecting lines) in which all three lines go through a single point, but the lines remain. I count them also. Walter Whiteley


