(Photograph:  Photograph: WWL-TV, from The Guardian)

Teenagers prove Pythagoras using Trigonometry

The Guardian recently reported that 2 US teenagers discovered a new proof for Pythagoras using trigonometry.  Whilst initial reports claimed incorrectly that this was the first time that Pythagoras had been proved by trigonometry, it is nevertheless an impressive achievement.  I will go through what I think the proof is below (this has not been officially published yet but was suggested by Mathtrain’s video here).

It’s worth noting that this proof is somewhat similar (and quite a lot less elegant!) than a proof by John Arioni which also makes use of a geometric progression of similar triangles – but still it’s an interesting exercise to work through.

The initial sketch:

We start with a right angled triangle ABC, such that angle ABC = beta  and angle BAC = alpha.  We label sides BC = a, CA = b and AB = c.  Next we draw a congruent triangle ACD.  We then draw the perpendicular line to AB through point B and we also extend the line AD until it meets this perpendicular at E.  From triangle ABC we can note that alpha + beta = 90 degrees .  This means that angle DBF = alpha  and therefore angle BFD = beta.  We can also note that ADG is a straight line which means that angle FDG = alpha, and therefore angle FGD = beta.  We can then note that all of these subsequent triangles share angles 90, alpha and beta and so are similar.  There will be an infinite number of these triangles formed as we approach point E.

First we define sin(alpha) and sin(beta) using triangle ABC:

Next we find the length BF using triangle BDF:

and we then use our earlier result to give:

Next we use this result for the hypotenuse of triangle BDF to find DF:

Next we find DG in triangle DFG:

We then compare the hypotenuse BF in triangle in BDF and the hypotenuse DG in triangle in DFG.  These are similar triangles and so the scale factor is given by:

This scale factor can be calculated in a similar manner when going from DFG to FGH as also being the same, and holding for subsequent triangles in this pattern.

Next we look at the relationship between BF and FH.  These are two hypotenuse of similar triangles calculated by first a scale factor from BF to FG and then the same scale factor from FG to FH.  Therefore the scale factor from BF to FH is:

Next we can say that BE is the sum of an infinite geometric sequence:  BF + FH + …. This allows us to use the sum of an infinite geometric sequence formula (we can define a<b in our original sketch to ensure that this converges).

By a similar method we say that AE is the sum of:  AD + DG + GI +…

Now going back to triangle ABE we have:

And lastly looking at triangle ABD we have:

Which we can rearrange to give:

And as if by magic, Pythagoras’ theorem appears!  It’s incredible to think that after over 2000 years, new proofs are still being discovered – and this one by two teenagers.   Maybe you too could discover a new proof?