Euler does an Almost Pythagorean Theorem

A while back I posted several times about relations that reminded me of the Pythagorean Theorem. After several requests I rushed out a short paper of the ones I could remember best.

Today I found another while reading one of the wonderful articles by Ed Sandifer that were regularly featured in the MAA, "How Euler Did It."
This particular one was is titled "Beyond Isosceles Triangles".

This is about Euler's paper E324 -- Proprietates triangulorum, quorum anguli certam inter se tenent rationem (Properties of triangles for which certain angles have a ratio between themselves) which is not yet translated at The Euler Archive.

"We know lots about triangles for which Angle A = Angle B. Such triangles are isosceles, and we have known at least since Euclid that Angle A = Angle B exactly when a = b. In 1765, Euler studied a generalization of this situation. What happens if Angle B is some multiple of Angle A ?"

The one that caught my eye says "if the sides of a triangle satisfy the relation ac = bb- aa , then Angle B = 2 Angle A ." ..
I prefer it better as a² + ac = b².

Students should know a couple of candidates in which one angle is twice another.  The 30, 60. 90 for example, and an isosceles right triangle should both be candidates to confirm. 



Euler goes on to prove that when Angle B is three times Angle A, then (b² -a²)(b - a) =ac² . Beautiful, but not quite so "Pythagorean"..

The geometry is clever, and probably clear enough for a really good high school geometry student to handle. Too many diagrams to copy here, so give it a read.
Sandifer says Euler continues through a ratio of 5 to 1, spots a pattern and extends the results to 13.

Sounds like a neat class project to me.