Let $$\small F_n=(a+b+c)^n+(b+c+d)^n-(c+d+a)^n-(d+a+b)^n+(a-d)^n-(b-c)^n$$ and $ad=bc$, then $$64 F_6 F_{10}=45 F_8^2$$ This fascinating identity is due to Ramanujan and can be found in http://www.maa.org/programs/maa-awards/writing-awards/ramanujan-for-lowbrows (Ramanujan for Lowbrows, by B.C. Berndt and S. Bhargava). Would anyone have any idea how Ramanujan discovered this identity?

The proofs of the identity offered so far at http://www.jstor.org/discover/2324305 (A Note on an Identity of Ramanujan, by T. S. Nanjundiah), http://www.jstor.org/discover/10.2307/2589526 (Two or Three Identities of Ramanujan, by M.D. Hirschhorn) and http://journals.cambridge.org/article_S0017089500008910 (A remarkable identity found in Ramanujan's third notebook, by B.C. Berndt and S. Bhargava) make the identity less mysterious, but how Ramanujan found the identity in the first place still remains a mystery. As Berndt and Bhargava remarked, it is also not clear whether this is an accidental, isolated result (along with the 3-7-5 counterpart discovered by Hirschhorn), or if there is some deeper theorem lurking behind it.