Question: Is it true that area of $\Delta_{OPQ}\leq$ area of $\Delta_{OPR}$+ area of $\Delta_{ORQ}$ for $P,Q,R$ on the unit sphere, where $\Delta_{OPQ}$ means a triangle on vertices $O, P$ and $Q$.
Here I am taking $P,Q,R$ as arbitrary distinct points on $\mathbb{S}:=\{\vec r\in\mathbb{R}^3:|\vec r|=1\}.$
I have not got much clues about such problem, I figured all three triangles will be isosceles and there are a few formulae of areas I have tried without luck. Could someone please provide a counterexample or perhaps hint as to how can I approach this problem?
Many thanks!