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Given a hyperbolic triangle $T$ and two points $p$ and $q$ in Poincare disk. Note that $p$ and $q$ are outside the triangle. If $p$ has shorter distances to the three vertices of $T$ than $q,$ can we claim that $p$ has shorter distances to all points inside $T$ than $q$?

If the answer is true, can we extend ths claim from a hyperbolic triangle to a hyperbolic convex hull?

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