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)

Some Tidbits

Some Tidbits:

Phi (the golden ratio) plays a very important role in Mathematics,
Sciences, Arts and Zome tool. Please take a look at
http://redsevenone.wordpress.com/
(and click on the link on the right side ZOME - the Answers Before you ask for Zome tool),
and http://en.wikipedia.org/wiki/Golden_ratio (for Phi and Golden Ratio).

If Pi day is celebrated on March 14 (3/14), don't you think Phi day
should be celebrated on 6th of January (1/6)

If Phi is 1.6180339887, then 1/Phi is 0.6180339887 - Do you know why? Because Phi satisfies the algebraic equation Phi^2 = Phi + 1.
If you divide this equation by Phi, you see the answer. This trick was
revealed to me by none other than my co-blogger TV Raman