If the point $\left({\mathrm{x}}_1+\mathrm{t}\left({\mathrm{x}}_2-{\mathrm{x}}_1\right),{\mathrm{y}}_1+\mathrm{t}\left({\mathrm{y}}_2-{\mathrm{y}}_1\right),{\mathrm{z}}_1+\mathrm{t}\left({\mathrm{z}}_2-{\mathrm{z}}_1\right)\right)$ divides the line segment joining  $\left({\mathrm{x}}_1,{\mathrm{y}}_1,{\mathrm{z}}_1\right)$and $\left({\mathrm{x}}_2,{\mathrm{y}}_2,{\mathrm{z}}_2\right)$ internally, then

$\begin{array}{l}\therefore {\mathrm{x}}_1+\mathrm{t}\left({\mathrm{x}}_2-{\mathrm{x}}_1\right)={\mathrm{x}}_1+{\mathrm{tx}}_2-{\mathrm{tx}}_1\\ =(1-\mathrm{t}){\mathrm{x}}_1+{\mathrm{tx}}_2\\ =\frac{{\mathrm{tx}}_2+(1-\mathrm{t}){\mathrm{x}}_1}{\mathrm{t}+(1-\mathrm{t})}\end{array}$
Given point divides line segment joining  $\left({\mathrm{x}}_1,{\mathrm{y}}_1,{\mathrm{z}}_1\right)\mathrm{\&}\left({\mathrm{x}}_2,{\mathrm{y}}_2,{\mathrm{z}}_2\right)$ in the ratio $\mathrm{t}:1-\mathrm{t}$  internally.