Let $\beta=[z_1, z_2]$ and $\gamma=[z_1, z_3]+[z_3, z_2]$, where $z_1, z_2$ and $z_3$ are distinct point in the complex plane. Assuming that $z_3$ lies on the line segment with endpoints $z_1$ and $z_2$, show that $\beta$ can be obtained from $\gamma$ by the piecewise smooth change of parameter $h:[0, 1]\rightarrow [0,2]$ given as follows: with $\lambda=|z_3-z_1|/|z_2-z_1|$, $h(s)=s/\lambda$ when $0\leq s\leq \lambda$ and $h(s)=(s+1-2\lambda)/(1-\lambda)$ when $\lambda \leq s\leq 1$.
I have arrived at the following $\gamma(t)=z_1+t(z_3-z_1)$ when $0\leq t\leq 1$ and $\gamma(t)=z_3+t(z_2-z_3)$ when $1\leq t\leq 2$ and so, as I have to prove that $\beta(t)=\gamma(h(t))$ then I take $t\in [0,1]$ and so I have two cases and in the two I get to $\gamma(h(t))=z_1+t\lambda(z_3-z_1)$ and $\gamma(h(t))=z_3+((t+1-2\lambda)/(1-\lambda))(z_2-z_3)$ but I do not know what else to do here. Could anyone help me please? Thank you very much.