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I am trying to understand a proof from the book "Metric Spaces of Non-Positive Curvature". In the $(3)\Rightarrow (1)$ direction, I can get why is the internal point varies continuously. It makes sense, but I am having trouble proving it. Here is the definitions then the theorem and right after, the proof, as in the book: enter image description here

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The proof of the relevant part:

I have tried to show somehow that by defining $f:[0,1]\to [y,z]$ by $$f(t)=(\chi _{\Delta _t} |_{[y,z]})^{-1} (o_{\Delta _t})$$ we will get that $f(t)$ gives us an internal point of $\Delta _t$ but I am not sure on why is it continues. I thought using the fact that $\chi _{\Delta _t} |_{[y,z]}$ is an isometry and that $c$ is continues. Will be happy for some help!

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    You should probably say what $T_\Delta$ is. From context I know what it is, but it is worth defining in the body of the question. – 2017-02-21

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This is more of a hint/plan of attack:

Think about the corresponding tripod/(degenerate tripod at $t=0$) for the triangle, and note the length of the three sides determine a tripod, and you should be able to prove that the length of the sides are changing continuously. Since the lengths of the sides are changing continuously you get a tripods $T(a_t,b_t,c_t)$, and you can prove that $a_t$ (or whatever side you are looking at) are changing continuous (formally you may find using Gromov product useful). From here you should have that the the internal point is varying continuously.