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Let $\mathbb{S}^2$ be the unit sphere and $d$ be the geodesic distance. For any three points $A,B,C\in \mathbb{S}^2$ and $0<\lambda<1$, let $A_{\lambda}$ and $B_{\lambda}$ be points on the geodesic paths $[A,C]$ and $[B,C]$ such that $d(A_{\lambda},C)=\lambda d(A,C)$ and $d(B_{\lambda},C)=\lambda d(B,C)$. Assume that $d(C,A)\le \pi/2, d(C,B)\le \pi/2$. It is geometrically clear that $d(A_{\lambda},B_{\lambda})\ge \lambda d(A,B)$. But proving it mathematically seems troublesome. Does anyone have a good proof or a direct reference?

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    If you are not afraid to use high powered machinery, this probably follows from one of the [comparison theorems in Riemannian geometry](http://www.amazon.com/Comparison-Theorems-Riemannian-Geometry-Publishing/dp/0821844172), using that $\mathbb{S}^2$ has non-negative curvature. But you are right that there probably is a simpler proof in this special case.2011-11-15
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    It would have made sense to link to [this related question](http://math.stackexchange.com/questions/81878/ratio-of-geodesic-segments-on-the-sphere) of yours.2011-11-15

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