Quandaries and Queries


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





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 an-2 + an-1 = an

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'.

  • it is the 'limit' of what we talk about as triangles (with very small errors off the line)
  • it comes up from standard constructions with something like GSP that when something defines a 'triangle', then dragging points around gives three collinear points as one case.
  • the object satisfies all the usual qualities and formulae which we use for triangles (such as sum of the angles, perimeter, area, three medians meet in a point, .... )
  • we like to use 'triangles' to think about trig functions - but the only way to handle angles of 0 and 90 degrees is to include triangles with two points on top of one another, and therefore three points on a line!

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
York University



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