Thread: Help proving the golden ratio is "hereditary".

Need a little help with this excercise:

"Show that the golden ratio is hereditary. For this, take a CeAB, which divides this segment according to the golden ratio and a DeAC, such as AD ≡ CB. All you need to do is show that AC/AD = AD/DC. Rember that AC^2 = AB.CB."

Thanks

Hello,

I am not sure what "AD ≡ CB" means. Can you elaborate on that?

We can assume that BC = 1. Then, by assumption, AC = φ (the golden ratio) and AD = 1. The fact that AC^2 = AB * CB means

φ^2 = φ + 1. (1)

Express x through φ and write what you need to prove, i.e., AC / AD = AD / DC, in terms of φ. You should get an equation equivalent to (1).

Note on notation. You can write C ∈ AB by copying and pasting the Unicode symbol (e.g., this Wikipedia page has many Unicode math symbols). The easiest way is to say "C in (or on?) AB." Normally, one can use LaTeX on this site, and then you can also write [tex]C\in AB[/tex].

Thank you very much