The most success I have ever had teaching proofs at secondary school level is with the Peaucellier–Lipkin linkage. The proof relies on nothing more than basic geometry, similar triangles, etc, but the outcome really is amazing. I found it reading Tom Korner's book Fourier Analysis, which must be one of the best books written about mathematics by a mathematician ever. You can get the proof from there or from Wikipedia, and there are some videos around if you google for them. Korner of course goes into the history of the problem, he believes Tchebychev was of the opinion that the problem couldn't be solved! And there is a wonderful quote from Kelvin, which I'll leave you with. When Sylvester showed a working model to him...

'[he] nursed it as if it had been his own child, and when a motion was made to relieve him of it, replied "No! I have not had nearly enough of it - it is the most beautiful thing I have ever seen in my life."'

The most success I have ever had teaching proofs at secondary school level is with the Peaucellier–Lipkin linkage. The proof relies on nothing more than basic geometry, namely similar triangles, but the outcome really is amazing. I found it reading Tom Körner's book called Fourier Analysis. 

You can get the proof from Wikipedia, and there are some videos around if you google for them. Körner goes into the history of the problem, he believes Tchebychev was of the opinion that the problem couldn't be solved! And there is a great quote attributed to Kelvin, which I'll leave you with. When Sylvester showed a working model to him...

'[he] nursed it as if it had been his own child, and when a motion was made to relieve him of it, replied "No! I have not had nearly enough of it - it is the most beautiful thing I have ever seen in my life."'