$AA_1$, $BB_1$, $CC_1$ are the medians of triangle $ABC$ whose centroid is $G$. If points $A, C_1, G, B_1$ are concylic then prove that $2a^2= b^2 + c^2$.
My try:- $ar(GBC)=1/3ar(ABC)$
Now I can't think any further. Here is my diagram: