Golden Ratio and Right Angle Triangle

I truly enjoy helping high school and middle school students reach their potentials in solving Mathematical Problems. In this process, I also learn and relearn.

I was working on a right angle triangle problem with some high school students. I realized that there are infinitely many triplets (a,b,c) of side lengths which are in arithmetic progression - for different increments - such as (3,4,5), or any integer multiple of this. We can also get rational increments - again an infinite number of rationals.

On the other hand, for (a,b,c) to be in geometric progression, the ratio can be only one - namely the square-root of (\phi) (or 1.2720196492 - up to 10 decimal places)