<s>[INST] in a triangle $ABC,$ if $2c^2=a^2+b^2,$ then largest possible degree measure of angle $C$ is In a triangle $ABC,$ if $2c^2=a^2+b^2,$ then largest possible degree measure of angle $C$ is 
cosine formula $\displaystyle \cos C = \frac{a^2+b^2-c^2}{2ab} = \frac{c^2}{2ab}<\frac{(a+b)^2}{2ab}=\frac{a^2+b^2}{2ab}+1$
wan,t be able to process after that, could some help me
 [/INST] We know that $$\cos C =\frac {a^2+b^2-c^2 }{2ab} =\frac {c^2}{2ab} =\frac {a^2+b^2}{4ab} $$ Now applying the AM-GM inequality, we get, $$\frac {a^2+b^2}{4ab}\geq \frac {1}{2} \Rightarrow C\leq \frac {\pi}{3} $$ Hope it helps. 
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